Automorphisms of free products of groups
A dissertation submitted to the
University of Cambridge for the
degree of Doctor of Philosophy
James T. Griffin
Corpus Christi College
Department of Pure Mathematics
and Mathematical Statistics
c© 2011 James T. Griffin
To my grandparents.
Abstract
The symmetric automorphism group of a free product is a group rich in algebraic structure
and with strong links to geometric configuration spaces. In this thesis I describe in detail
and for the first time the (co)homology of the symmetric automorphism groups.
To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism
group, a large normal subgroup of the symmetric automorphism group. This classifying
space is a moduli space of ‘cactus products’, each of which has the homotopy type of a
wedge product of spaces.
To study this space we build a combinatorial theory centred around ‘diagonal com-
plexes’ which may be of independent interest. The diagonal complex associated to the
cactus products consists of the set of forest posets, which in turn characterise the homol-
ogy of the moduli spaces of cactus products. The machinery of diagonal complexes is
then turned towards the symmetric automorphism groups of a graph product of groups.
I also show that symmetric automorphisms may be determined by their categorical
properties and that they are in particular characteristic of the free product functor. This
goes some way to explain their occurence in a range of situations.
The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres
embedded disjointly in (n+ 2)-space. When n = 1 this is the configuration space of
unknotted, unlinked loops in 3-space, which has been well studied. We continue this work
for higher n and find that the fundamental groups remain unchanged. We then consider
the homology and the higher homotopy groups of the configuration spaces.
Our last contribution is an epilogue which discusses the place of these groups in the
wider field of mathematics. It is the functoriality which is important here and using this
new-found emphasis we argue that there should exist a generalised version of the material
from the final chapter which would apply to a far wider range of configuration spaces.
i
ii
Declaration
This dissertation is the result of my own work and includes nothing which is the outcome
of work done in collaboration except where specifically indicated in the text. No part
of this dissertation is substantially the same as any that I have submitted or will be
submitting for a degree or diploma or other qualification at this or any other University.
James Thomas Griffin
Cambridge, August 2011.
Parts of this work are presented in the following article,
• J.T.Griffin. Diagonal complexes and the integral homology of the automorphism
group of a free product. Submitted.
iii
iv
Acknowledgements
This work was carried out while the author was funded by the Richard Metheringham
Scholarship, awarded by the Worshipful Company of Cutlers, I am very grateful for both
their funding and their hospitality. The department in Cambridge is a friendly and in-
spiring place and I would like to thank everyone who has helped me over the years. In
particular I am grateful
• to Christopher Brookes my PhD superviser for all of his guidance,
• to Martin Hyland for his advice and insight,
• to Peter LeFanu-Lumsdaine and Richard Garner for all the conversations concerning
category theory,
• to Simon Wadsley, Chris Bowman, Rishi Vyas, Jonathon Nelson and the rest of the
algebra group for all of their algebra related discussions
• and to Marj Batchelor, who has had such a huge influence on the way I think about
mathematical communication, learning and teaching.
From further afield Vladimir Dotsenko introduced me to the groups which are at the
centre of my thesis and we have had many great conversations about operads. Aure´lien
Djament and Gae¨l Collinet provided feedback on my paper and were kind to invite me to
Nantes for a week in February 2011.
Finally thank you to my family; my parents and of course Juliette, whose support has
been so valuable.
v
vi
Contents
Introduction 1
1 Preliminaries 7
1.1 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 The Seifert-van Kampen theorem . . . . . . . . . . . . . . . . . . . 8
1.1.2 Colimits of CW complexes . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.3 Homotopy quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Graph products and right-angled Artin groups . . . . . . . . . . . . . . . . 11
1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Right-angled Artin groups . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 A topological approach . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.4 Flag complexes and classifying spaces . . . . . . . . . . . . . . . . . 16
1.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Diagonal complexes 21
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Diagonal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 The poset of partial partitions . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Diagonal complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.4 The geometric realisation . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.6 Morphisms of diagonal complexes . . . . . . . . . . . . . . . . . . . 30
2.1.7 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Products of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 As a fundamental group . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 An explicit presentation . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 (Co)homology of products . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 (Co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Structure of the cohomology ring . . . . . . . . . . . . . . . . . . . 43
2.3.3 Extensions by symmetry groups . . . . . . . . . . . . . . . . . . . . 44
2.3.4 Homotopy quotients by automorphism groups . . . . . . . . . . . . 45
vii
viii CONTENTS
2.4 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.1 Orthogonal decomposition . . . . . . . . . . . . . . . . . . . . . . . 48
2.4.2 Conical decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Coset complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6 Questions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3 Symmetric automorphisms of free products 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Moduli spaces of cactus products . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Cactus products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 The moduli space of cactus products . . . . . . . . . . . . . . . . . 60
3.2.3 An embedding into a direct product . . . . . . . . . . . . . . . . . . 61
3.2.4 Geometric interpretation of partial conjugations . . . . . . . . . . . 62
3.3 The diagonal complex of forest posets . . . . . . . . . . . . . . . . . . . . . 63
3.3.1 The construction of ΓFn . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.2 Properties of YΓFn . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.3 Decompositions of ΓFn . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.4 McCullough-Miller space and coset complexes . . . . . . . . . . . . 71
3.3.5 Based McCullough-Miller space . . . . . . . . . . . . . . . . . . . . 74
3.4 (Co)homology of the symmetric automorphism groups . . . . . . . . . . . . 75
3.4.1 The Fouxe-Rabinovitch groups . . . . . . . . . . . . . . . . . . . . . 75
3.4.2 The pure automorphism groups . . . . . . . . . . . . . . . . . . . . 78
3.4.3 The symmetric automorphism groups . . . . . . . . . . . . . . . . . 79
3.5 Categorical interpretation of symmetric automorphisms . . . . . . . . . . . 81
3.5.1 Extendable automorphisms . . . . . . . . . . . . . . . . . . . . . . 82
3.5.2 A characterisation of WH(G) . . . . . . . . . . . . . . . . . . . . . 83
3.6 Relationship with Outer space . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.6.1 Background on Outer space . . . . . . . . . . . . . . . . . . . . . . 85
3.6.2 Cactus graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6.3 Outer space with boundaries . . . . . . . . . . . . . . . . . . . . . . 86
3.7 Automorphisms of graph products . . . . . . . . . . . . . . . . . . . . . . . 89
3.7.1 The diagonal complex ΓG . . . . . . . . . . . . . . . . . . . . . . . 89
3.7.2 Action on the graph products . . . . . . . . . . . . . . . . . . . . . 93
4 Configuration spaces 97
4.1 The spaces of codimension 2 spheres . . . . . . . . . . . . . . . . . . . . . 98
4.1.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.2 The structure of the configuration space . . . . . . . . . . . . . . . 100
4.1.3 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Other homotopy invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
CONTENTS ix
4.2.1 The homology groups of Ln(k) . . . . . . . . . . . . . . . . . . . . . 105
4.2.2 The configuration space when k = 2 . . . . . . . . . . . . . . . . . . 106
4.2.3 The suspension ΣLn(k) . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.4 Remarks on the higher homotopy groups . . . . . . . . . . . . . . . 108
4.2.5 Another classifying space for FR(Fk) . . . . . . . . . . . . . . . . . 109
4.2.6 Sketch of a possible proof of Conjecture 4.2.1 . . . . . . . . . . . . 110
Epilogue 113
References 122
x CONTENTS
Introduction
The free product of a pair of groups G and H is easy to describe; one takes a presentation
〈X | R〉 for G and a presentation 〈Y | S〉 for H and then takes their union 〈X ∪ Y | R ∪ S〉
which is a presentation for the free product G ∗H. Equivalently the free product may be
described using the language of category theory as the binary coproduct in the category
of groups. For groups G and H the free product is the unique (up to isomorphism) group
G ∗H such that for any group K with maps G → K and H → K there exists a unique
factorisation
G
##G
GG
GG
GG
GG H
{{ww
ww
ww
ww
w
G ∗H
K
The n-ary free product of groups G1, . . . , Gn is given by repeated applications of the binary
free product which is associative. These objects occur naturally. Suppose one has a pair
of connected, pointed spaces X and Y , then the wedge product is given by identifying
their basepoints. The Seifert van-Kampen theorem tells us that the fundamental group
of the wedge product is the free product of the fundamental groups of X and Y .
So free products are simple to describe and occur naturally. It turns out that they
also have an interesting group of symmetries.
Automorphisms of free products
A presentation for the automorphism group of a free product of groups has been known
for some time [21]. Suppose that we are given a group as a free product G = G1 ∗ . . .∗Gn;
to give a presentation of the automorphism group, Aut(G) we would need to know
A. the automorphism group Aut(Gi) of each individual factor,
B. which of the pairs Gi, Gj are isomorphic,
C. which of the Gi are infinite cyclic, and finally
D. that each Gi is freely indecomposable.
1
2 INTRODUCTION
If we knew A, B, C and D then we would be able to give the presentation. So what does
each individual piece of information give us?
To write down the presentation we need to know A, B and C, then the final piece D
would tell us that the group obtained is in fact the full automorphism group. If we were
to find that D were false then the group written down would still act on the free product,
but it would be a proper subgroup of Aut(G). Similarly if we know only a subset of A, B
and C then we could still write down a partial presentation, however the group presented
would be a subgroup of Aut(G).
The group obtained when only A and B are taken into account is called the symmetric
automorphism group, ΣAut(G) and it is this group which is the central object of study
in this thesis. It is the full automorphism group when the answer to C is that no Gi is
infinite cyclic and when D is known to be true. The main step in studying ΣAut(G) is
to first understand the group of automorphisms we may give when we know none of A,
B, C or D; this group is called the Fouxe-Rabinovitch group, FR(G). The presentation of
FR(G) contains only the relations from the factor groups Gi and certain commutators.
Viewed from this perspective the Fouxe-Rabinovitch group does not depend on any of the
properties of the factors and so it should come as no surprise that it is (a) functorial in
the factor groups and (b) very well behaved.
History
As we have already mentioned, a presentation of Aut(G1 ∗ . . . ∗ Gn) where A, B, C and
D are known was given by Fouxe-Rabinovitch [21]. This also gives a presentation of the
symmetric automorphism group ΣAut(G1 ∗ . . . ∗Gn) for any Gi (we merely pretend that
each Gi is freely indecomposable and not infinite cyclic). The group ΣAut(Z∗n) appeared
in the study of configuration spaces as the fundamental group of the configuration space
of n unlinked, unknotted loops in 3-space [18]. A refined presentation of ΣAut(Z∗n) was
given by McCool [36]. It was not long after that significant progress was made on the
cohomological properties of this group [14]. Collins described a version of Outer space
consisting of graphs in a cactus shape; these may be seen as prototypical examples of the
cactus products introduced in this thesis.
Attention next moved to the cohomology groups of ΣAut(Z∗n); the integral cohomol-
ogy of FR(Z∗3) was calculated in [9] and Brownstein and Lee conjectured the structure for
general n. It should be stressed that up to this point most attention had been restricted
to the case of Z∗n rather than general free products.
In [37], McCullough and Miller described a space on which ΣAut(G1 ∗ . . . ∗Gn) acts.
In particular they showed that the space is contractible and that the action of FR(G) has
stabilisers which are direct products of the Gi. This was however a departure from the
space described by Collins [14]. Still the space allowed many results to be established [5],
[35], [13] culminating in a proof of the Brownstein-Lee conjecture [27] for ΣAut(Z∗n).
INTRODUCTION 3
Since then calculations of the Euler characteristic [28] of ΣAut(G) and partial calculations
of the cohomology of ΣAut(G) over a field have appeared [4], see Remark 3.4.10 for a
discussion.
The Outer space of a free group
To understand the purpose of McCullough-Miller space it is best to first look at the
motivation from the theory of automorphism groups of free groups. Let Fn be the free
group on n letters, this is the group one obtains when given n generators and no relations.
In one sense this is one of the simplest groups, for example to give a homomorphism from
Fn to a group H it is enough to list the elements of H to which each generator of Fn
is assigned; since there are no relations to check this is all that is required. Despite
this the symmetries of the free group Fn are rather complicated and the study of the
automorphism group Aut(Fn) has been long, there are many open problems and much is
left to explore.
To begin with algebraic and combinatorial techniques were used to study Aut(Fn)
and these yielded a presentation. In this thesis we will not interest ourselves with such
methods, we will concentrate on the topological methods. The breakthrough in this area
was made in [17] with the definition of the ‘Outer space’ of a free group, although the
name came later. The basic idea is that since a rank n graph has fundamental group a
free group Fn and since graphs can be continuously altered by adjusting edge lengths and
merging and splitting vertices, a path in the ‘moduli space of rank n graphs’ will trace
out an automorphism of the free group Fn.
Culler and Vogtmann made this intuition precise and by looking at ‘marked graphs’,
which are graphs decorated with a basis of Fn, one may define a moduli space which
holds an action of Aut(Fn). Furthermore they showed that this space was contractible
and although the action by Aut(Fn) wasn’t free, the simplex stabilisers are finite. So
the motivation behind McCullough-Miller space is to provide a space on which Aut(G1 ∗
. . . ∗ Gn) acts. They were successful and defined a contractible space and although the
stabiliser subgroups are not necessarily finite, they are well understood in terms of the
factor groups G1, . . . , Gn. Outer space and McCullough-Miller space are combined in the
Outer space of a free product defined in [23].
4 INTRODUCTION
A different approach
The methods used in this thesis are different in that the object at the centre is not
McCullough-Miller space, but we are still motivated by ‘Outer space’, we just take things
in a slightly different direction. The approach is more in line with that of Collins [14]. The
central construction is a functor that takes an n-tuple of pointed spaces Y = (Y1, . . . , Yn)
and gives a moduli space MY of objects which we name cactus products. An example
of a single cactus product is pictured below.
Y4
Y5
Y2
Y3
Y1
A cactus product of pointed spaces is much like a wedge product, in fact it is homotopy
equivalent to the wedge product. The space of cactus products is naturally functorial and
there is an equivalence of functors
pi1(MY) ∼= FR
(
pi1(Y1) ∗ . . . ∗ pi1(Yn)
)
,
which says that the fundamental group of the moduli space of cactus products is the
Fouxe-Rabinovitch automorphism group of the free product pi1(Y1) ∗ . . . ∗ pi1(Yn).
A stronger result holds.
Theorem. Suppose that Y1, . . . , Yn are classifying spaces for G1, . . . , Gn. Then MY is a
classifying space for FR(G1 ∗ . . . ∗Gn).
This is the central theorem of this thesis and it is proved in Section 3.3.
Diagonal complexes
To prove this we require a strong handle on the moduli space and so an alternative
description is required: we construct it as a diagonal complex product. When introducing
the notion of diagonal complex we are inspired by graph products, indeed the diagonal
complex products are generalisations of graph products. Recall that for each graph Γ
with vertex set [n] there is a product of groups G = (G1, . . . , Gn) which is generated
by the groups Gi along with addition relations: for each edge (i, j) ∈ E(Γ) we have the
commutator relations [gi, gj] for gi ∈ Gi and gj ∈ Gj. Similarly for each diagonal complex
there will be defined a product of groups, although the presentation for the products is
necessarily more complicated.
INTRODUCTION 5
We will see that there is a diagonal complex of forest posets which corresponds to
the Fouxe-Rabinovitch automorphism groups. This diagonal complex will also define a
product of pointed spaces which will give the moduli space of cactus products. It is
the formalism of diagonal complexes which allows us to prove the theorem above using
combinatorial properties of the forest posets and a variant of McCullough-Miller space
from [13]. The diagonal complex also naturally describes the (co)homology of the moduli
space and by extension the (co)homology of both the groups FR(G) and the symmetric
automorphism groups ΣAut(G).
Further applications
Other diagonal complexes describe other products of groups. There is a generalisation
of a forest poset for each graph G which we call a G-admissable poset; a forest poset
is G-admissable for the discrete graph (consisting only of vertices). The G-admissable
posets also define diagonal complexes and we conjecture that the corresponding product
of groups H = (H1, . . . , Hn) is the symmetric automorphism group of the graph product
HG.
One of the motivations for research into the symmetric automorphism group of a
free group Fn ∼= Z∗n was that it is the fundamental group of the configuration space of
n unknotted, unlinked loops in R3. As such we investigate the configuration spaces of
k-spheres in Rk+2. In fact the fundamental groups are isomorphic to FR(Z∗k) for all
k ≥ 1.
Plan of the thesis
In Chapter 1 we review graph products of groups and their (co)homological properties.
The graph product GΓ of factor groups G = (G1, . . . , Gn) over a graph Γ is functorial
with respect to the factor groups and we proceed by defining classifying spaces for the GΓ.
Ofcourse we do this functorially obtaining spaces YΓ for pointed spaces Y = (Y1, . . . , Yn).
The purpose of this chapter is to introduce the general methodology we use for the more
complicated functors which follow.
So in Chapter 2 we introduce diagonal complexes, which are more versatile than graph
products. We give formulae for the (co)homology of a diagonal complex product of pointed
spaces as well as presentations of the diagonal complex products of groups. Unfortunately
the extra versatility comes at a price and it is now harder to determine whether a space is
a classifying space or not. The chapter finishes with two decomposition theorems which
are aimed at making this task easier along with the definition of a coset complex which
allows geometric techniques to be used.
It is in Chapter 3 that we turn to symmetric automorphisms. Cactus products and
their moduli spaces are introduced and the geometric interpretation of partial conjugations
6 INTRODUCTION
is described. The commutator relations between partial conjugations may be seen as tori
embedded in the moduli space. The purpose of Chapters 1 and 2 was to give a gentle
introduction to diagonal complexes and it is here that we give their first major application.
We prove that the set of forest posets is in fact a diagonal complex and furthermore that
it does give the moduli spaces of cactus products and the groups of partial conjugations
that we wanted to study. Now that we have a good description of the moduli spaces
we can prove that they are classifying spaces for the groups FR(G1 ∗ . . . ∗ Gn) when
the factor spaces are classifying spaces for the Gi. The formulae for the (co)homology of
diagonal complexes, established in Chapter 2 now give the (co)homology of the symmetric
automorphism groups.
We then go on to discuss the relationship between Outer space and the moduli space
of cactus products, in particular we define a hybrid space which we conjecture to be
aspherical. If this conjecture were to be true it would be reasonable to call the space an
Outer space of a free product, the advantage over the definition of [23] is that the stabiliser
subgroups are far smaller. We also turn our attention to symmetric automorphisms of
graph products of groups, which involves new diagonal complexes and significantly more
complicated combinatorics.
The final Chapter 4 is devoted to configuration spaces. The configuration space of
n unknotted, unlinked loops embedded in 3-space is well treated in the literature. We
look at the analogue for k-spheres embedded in (k + 2)-space and find that they have
the same fundamental group FR(Z∗n). We then turn to the higher homotopy groups and
conjecture that for j ≥ 2 the homotopy groups pij vanish for k ≥ j. This would mean
that when one takes the limit as k → ∞ one gets a classifying space for FR(Z∗n). The
main distinction between this classifying space and the moduli space of cactus graphs is
that the action of the symmetric group is proper.
We finish the thesis with a short Epilogue. The purpose of this is to try to bring
together the different strands that run through the thesis. In particular we emphasise
the importance of the functoriality of our constructions. Ofcourse there is one area that
lacks any functoriality at all, that of the configuration spaces in the final chapter. In the
Epilogue we address this by making conjectures about more general configuration spaces
and sketch how the theory may pan out.
Chapter 1
Preliminaries
The purpose of this chapter is to provide a gentle introduction to the ideas contained
within the thesis. The diagonal complexes which are the subject of Chapter 2 offer a
generalisation of graph products of groups. Since our techniques simplify when applied to
graph products it makes sense to study this simplification before moving on to the general
case.
In Section 1.2 we do just that; we give a gentle introduction to graph products of
groups, we construct classifying spaces for these groups and then we calculate their
(co)homology. If the reader is already familiar with graph products then they may skip
this chapter without worry.
1.1 Algebraic topology
Algebraic topology is concerned with the properties and invariants of spaces up to homo-
topy. Let X be a path connected topological space with a chosen point p ∈ X. One of the
basic homotopy invariants of this space is the fundamental group pi1(X, p). But there are
other homotopy invariants, for instance the higher homotopy groups and the homology
and cohomology of the space.
We wish to study invariants of a group G. The general idea is to construct a classifying
space; a space X with fundamental group G and trivial higher homotopy groups, and then
to study the homotopy invariants of this space.
It is desirable to describe small models of classifying spaces for a group. There are a
number of approaches to this. A common approach is to construct a contractible space
equipped with a proper action of the group, often the space will carry some kind of
geometry or Morse function allowing one to show that it is contractible. The quotient of
the space by the group action is then a classifying space for the group.
Another method is to show that the classifying spaces are gluings of pairs of nice sub-
spaces which are themselves classifying spaces of subgroups. One of the requirements for
this is that the inclusion of the intersection of the subspaces into the subspaces themselves
7
8 1. PRELIMINARIES
must be injective on fundamental groups. The theorem required for this is the Seifert-van
Kampen theorem which is reviewed below. It is this method that we will apply to the
graph products of groups described later in this chapter.
For our central theorem in Chapter 3 we will actually use a combination of these two
approaches.
1.1.1 The Seifert-van Kampen theorem
CW complexes
In this thesis all spaces will be assumed to be CW complexes and all maps between
spaces will be maps of CW complexes. For definitions and a detailed treatment of their
properties see [25]. The reason for considering CW complexes is that we avoid pathological
examples of topological spaces whilst (for the purposes of algebraic topology) retaining
full generality. In fact every topological space X can be approximated by a CW complex
X ′ in such a way that their homotopy groups are identical, we say that X and X ′ are
weakly homotopic. This means that all the algebraic invariants we use see X and X ′ as
being identical.
We must be a little careful with our CW complexes. In the category of CW complexes
the colimit of a diagram is homeomorphic to the colimit in the category of topological
spaces, however the limit of a diagram in the category of CW complexes is not necessarily
homeomorphic to the limit in topological spaces. The limits have isomorphic underlying
sets but the topologies differ slightly, although (importantly) the two limits are weakly
homotopic. In practice the only limit we take in this thesis is the product. This means
that when we take the direct product of two CW complexes we don’t endow it with the
product topology, but with a slightly weaker topology, the CW product topology.
The important fact to take from the above comments is that if we start with CW
complexes, all of our constructions will still be CW complexes, so we will be able to apply
Theorem 1.1.1 and Proposition 1.1.2 without worry.
The Seifert-van Kampen theorem
The following theorem can be found in [7]. It may also be viewed in the context of
Bass-Serre theory.
Theorem 1.1.1. Suppose that X is a CW complex with subcomplexes A and B so that
X = A ∪ B. Suppose further that X, A, B and C := A ∩ B are connected. By picking a
point in the intersection C these spaces may be pointed. Then the fundamental group is
an amalgamation of the fundamental groups of the spaces A, B and C:
pi1(X) ∼= pi1(A) ∗pi1(C) pi1(B). (1.1.1.1)
1.1. ALGEBRAIC TOPOLOGY 9
Now suppose further that A, B and C are classifying spaces for pi1(A), pi1(B) and pi(C)
respectively. Suppose also that both pi1(C) → pi1(A) and pi1(C) → pi1(B) are injective.
Then X is a classifying space for pi1(X).
The theorem allows us to construct classifying spaces for amalgamations.
Example 1.1.1.1. Let G be the group with presentation
〈a, b, c | [a, b] , [b, c]〉 . (1.1.1.2)
The subgroups 〈a, b〉 and 〈b, c〉 are both isomorphic to Z2. Their intersection is 〈b〉 ∼= Z
and G is equal to the amalgamation
〈a, b〉 ∗〈b=b′〉 〈b′, c〉 . (1.1.1.3)
Let X be the space given by glueing two tori A,B together along radial circles.
(1.1.1.4)
Then applying the theorem shows that X is a classifying space for G. We may use this
to calculate the homology of G with coefficients in any ring R.
Hi(G,R) = Hi(X,R) =
0 if i ≥ 3,
R2 if i = 2,
R3 if i = 1 and
R if i = 0.
(1.1.1.5)
1.1.2 Colimits of CW complexes
One of the reasons for restricting ourselves to CW complexes, besides being a convenient
setup for the Seifert-van Kampen theorem, is that a CW complex may be reconstructed
from a covering of CW subcomplexes in the following sense.
Let X be a CW complex and let {Xi | i = 1, . . . , n} be CW subcomplexes of X. Then
if each point p ∈ X is in some Xj we say that {Xi} is a CW cover of X. The intersection
of two CW subcomplexes Xi ∩Xj is still a CW subcomplex and hence so is
⋂
i∈AXi for
10 1. PRELIMINARIES
each A ⊆ [n]. For each CW cover {Xi} of a CW complex X there is a diagram D in CW
complexes which has
• objects ⋂i∈AXi for each A ⊆ [n] and
• morphisms the natural inclusions ⋂j∈BXj ↪→ ⋂i∈AXi for each A ⊆ B ⊆ [n].
Proposition 1.1.2. Let X be a CW complex and {Xi | i = 1, . . . , n} be a CW cover. Let
D be the natural diagram associated to {Xi}. Then the colimit of D is isomorphic to X.
See [25], Section 4.G for a discussion of this result.
1.1.3 Homotopy quotients
Let G be a group acting on a space Y . The homotopy quotient can be calculated as
follows:
• take a contractible space E on which G acts properly,
• now consider the product E × Y on which G acts diagonally,
• we may take the quotient by the group action
E ×G Y = E × Y
(e, y) ∼ (g.e, g.y) | g ∈ G (1.1.3.1)
and since G acts properly on E × Y this is the homotopy quotient.
We now treat the analogue for homology. Let M be a differentially graded (d.g.)
RG-module, and suppose also that M is projective over R. The homotopy quotient, or
homology of G with coefficients in M , is calculated as follows:
• take an RG-projective resolution of the trivially RG-module R
. . .→ P2 → P1 → P0 → R→ 0. (1.1.3.2)
An example is given by C∗(E), where E is as above.
• Take the tensor product P• ⊗M on which RG acts diagonally,
• now take the quotient (P• ⊗M)G.
The connection between the homotopy quotient of a G-space and the homological quotient
is given by
H∗(E ×G Y,R) ∼= H∗
(
(C∗(E)⊗ C∗(Y ))G , R
)
. (1.1.3.3)
So the homology of E ×G Y is the homology of (P• ⊗ M)G when M = C∗(Y ) and
P• = C∗(E).
1.2. GRAPH PRODUCTS AND RIGHT-ANGLED ARTIN GROUPS 11
Finally we recall that if Y is a K(H, 1) with an action of G, then E ×G Y is a
K(H o G, 1). This means that the homology of H o G with coefficients in R is the
homology of the G-module C∗(Y )
H∗(G,C∗(Y )). (1.1.3.4)
A note of warning, the homology of H, which is given by H∗(C∗(Y )) is a G-module,
but there is not necessarily a G-module map to C∗(Y ). This means that the homology
H∗(G,H∗(H)) does not necessarily compute the homology of H o G. It is the first page
of a spectral sequence which computes H o G, however we will not need to use spectral
sequences if we are careful about taking homology.
1.2 Graph products and right-angled Artin groups
The graph product of groups assigns to a graph Γ = (V,E) and a V -labelled set of groups
G the graph product GΓ. When the graph has no edges the product is the free product
of groups and when the graph is complete it is the direct product. So a graph product
may be seen as an interpolation between the free and direct products. When each group
is infinite cyclic the product ZΓ is called a right-angled Artin group (RAAG) and is often
denoted AΓ. The name ‘right-angled Artin group’ comes from the fact that AΓ can be
given as the Artin group of a Coxeter matrix (mij) where each mij is either 2 or ∞.
The RAAGs were studied before the graph products, an early reference is [3]. Much
has been written since, for a summary with a long list of references see [10]. The general
graph products were studied in the thesis [22], where a normal form for group elements
was given.
1.2.1 Definitions
A graph Γ = (V,E) consists of a set of vertices V and a set of edges E consisting of
2-elements subsets of V . If e = {v1, v2} ∈ E then we say that v1 and v2 are the ends of
the edge e.
Important examples are the discrete graph DV on a vertex set V , where E is empty
and the complete graph KV on a vertex set V , where E is the set of all 2-elements subsets
of V .
Definition 1.2.1. Let Γ = (V,E) be a graph and let G be a V -tuple of groups, that is a
V -indexed set of groups {Gv}v∈V . Then the graph product of G indexed by Γ is the group
GΓ := 〈Gv | R〉 , (1.2.1.1)
12 1. PRELIMINARIES
where v ranges over V and R is the set of relations
R :=
⋃
e∈E
{
[g, h] | g ∈ Gv1 and h ∈ Gv2 , where e = {v1, v2}
}
(1.2.1.2)
We will refer to the groupsGv as vertex groups. So a graph product is a group generated
by the vertex groups, the additional relations assert that two elements of distinct vertex
groups commute when the vertices are joined by an edge.
Example 1.2.1.1. Let DV be the discrete graph on a vertex set V and let G be a V -tuple
of groups. Then GDV is the group generated by the vertex groups with no additional
relations (because DV has no edges). Hence GDV is the free product of the vertex groups.
Example 1.2.1.2. Let KV be the complete graph on a vertex set V and let G be a V -
tuple of groups. Then GKV is the group generated by the vertex groups where elements
from distinct vertex groups commute (because KV contains every possible edge). Hence
GKV is direct product of the vertex groups.
Example 1.2.1.3. Let Γ be the graph with V = {1, 2, 3, 4} and E = {12, 23, 34, 14},
where we write vw as a shorthand for the set {v, w}. This may be drawn as
• •
••
Let G = (G1, G2, G3, G4) be a 4-tuple of groups. Then the elements G1 in GΓ commute
with those of G2 and G4, but not with the non-trivial elements of G3. In fact we may
check that GΓ is isomorphic to
(G1 ∗G3)× (G2 ∗G4), (1.2.1.3)
the direct product of the free products of opposite vertex groups.
It is important to note that not every graph product may be written as a combination
of free and direct products. The smallest counterexample is given by:
Example 1.2.1.4. Let V = {1, 2, 3, 4} and let E = {12, 23, 13, 14} and E ′ = {12, 23, 14},
which define two graphs drawn as
• •
••
??????? &
• •
••
It is a fun and helpful exercise to check that the graph products indexed by these graphs
can not be given by iterated free and direct products. It is also worth checking that these
are the only such graphs with 4 vertices and that there are no such graphs with less than
4 vertices.
1.2. GRAPH PRODUCTS AND RIGHT-ANGLED ARTIN GROUPS 13
1.2.2 Right-angled Artin groups
Let Γ = (V,E) be a graph and let G be the V -tuple with each Gv = Z. Then GΓ
is the right-angled Artin group (or RAAG) indexed by Γ. It is often denoted AΓ. The
presentation of AΓ is relatively simple:
AΓ := 〈gv for v ∈ V | [gv1 , gv2 ] with v1, v2 ∈ E〉 . (1.2.2.1)
Remark 1.2.2. A morphism of graphs is defined by a map f on the set of vertices for which
each edge vw is either taken to an edge f(v)f(w) or f(v) = f(w). With this definition the
RAAG construction becomes a functor A• from the category of graphs to the category of
groups. This functor is a left adjoint, where the right adjoint Γ• is the following functor:
for a group G let ΓG be the graph with vertex set G and with edges the pairs gh for which
g commutes with h. The fact that these functors form an adjoint pair means that for
each graph H and each group G the two following homsets are naturally isomorphic:
Homgroups(AH , G) ∼= Homgraphs(H,ΓG). (1.2.2.2)
1.2.3 A topological approach
There is a similar product of pointed spaces. Rather than use graphs as indexing objects
we use simplicial complexes. A simplicial complex X = (V, S) consists of a set of vertices
V and a set of simplices S consisting of (finite) subsets of V . The set S must satisfy
another condition: if A ∈ S and B is a subset of A, then B ∈ S.
An example is the simplicial complex ∆V = (V, PfV ) consisting of the finite subsets
of V .
Example 1.2.3.1. Let Γ = (V,E) be a graph, then two natural constructions of simplicial
complexes are given by:
• let XΓ = (V, V ∪ E) be the simplicial complex consisting of the vertices and edges
of Γ.
• let FlΓ = (V, S) be the simplicial complex where S consists of the subsets A ⊆ V
such that each 2-element subset of A is in E. Alternatively S is the set of subsets A
such that the subgraph of Γ spanned by A is the complete graph KA. The simplicial
complex FlΓ is often called the flag complex of Γ.
One may also go the other way and construct a graph from any simplicial complex
X = (V, S). Let W = {v ∈ V | {v} ∈ S} and let E = S ∩ P2W consist of the 2-element
subsets in S. Then ΓX = (W,E) is the 1-skeleton graph of X.
14 1. PRELIMINARIES
Remark 1.2.3. The sets of graphs and of simplicial complexes actually form categories.
The morphisms are the maps on the underlying vertex sets which induce maps on the
edge and simplex sets respectively. In the example above we are actually describing two
functors X and Fl from the category of graphs to the category of simplicial complexes
and we describe one functor Γ going the other way. In fact Γ is left adjoint to Fl and
right adjoint to X.
Recall that a pointed space (P, ∗) is a topological space P equipped with a chosen
point ∗ ∈ P .
Definition 1.2.4. Let X = (V, S) be a simplicial complex and let P be a V -tuple of
pointed spaces {(Pv, ∗v)}v∈V . If p = (pv)v∈V ∈
∏
v∈V Pv then the support of p is the set
supp(p) := {v | pv 6= ∗v}. (1.2.3.1)
The simplicial complex product of P indexed by X is the subset
{p | supp(p) ∈ S} ⊆
∏
v∈V
Pv (1.2.3.2)
and is denoted PX. It is the set of points with support in S.
Example 1.2.3.2. Let V be a set and ∆V be the simplicial complex defined above and
let P be a V -tuple of pointed spaces. Then P∆V is the subspace of the direct product∏
v∈V Pv consisting of the elements with finite support.
Example 1.2.3.3. Recall that we constructed a space by gluing together two tori in
Example 1.1.1.1. This may be seen to be a simplicial complex product. Let X = (V, S)
be the simplicial complex with V = {1, 2, 3} and S = {1, 2, 3, 12, 23} which may be
visualised as
1 2 3 (1.2.3.3)
Then the space from (1.1.1.4) is isomorphic to the simplicial complex product S1X.
The simplicial complex product and graph product are closely linked. This is demon-
strated by
Theorem 1.2.5. Let X = (V, S) be a simplicial complex and let P be a V -tuple of pointed
CW complexes. Let G = {pi1(Pv)} be the V -tuple of their fundamental groups. Then the
fundamental group of PX is isomorphic to the graph product GΓX .
This theorem follows from the following two lemmas.
Lemma 1.2.6. Let X = (V, S) be a simplicial complex and let P be a V -tuple of pointed
CW complexes. Then PX is the colimit of the diagram which
1.2. GRAPH PRODUCTS AND RIGHT-ANGLED ARTIN GROUPS 15
• has objects the spaces PA :=
∏
v∈A Pv for A ∈ S, and
• has morphisms the inclusions
φ : PB ↪→ PA (1.2.3.4)
for each B ⊆ A ∈ S, where φ(p)v = pv if v ∈ B and φ(p)w = ∗w if w /∈ B.
Proof. Let A ∈ S, then the image of
PA =
∏
v∈A
Pv ↪→
∏
v∈V
Pv (1.2.3.5)
consists of all the elements which have support in either A or one of its subsets. The
simplicial complex product PX was defined as a subspace of the direct product
∏
v∈V Pv.
It consisted of the points p with support set contained in S. Hence all of the maps of
the form (1.2.3.5) form a cover of PX. Since each Pi is a CW complex and so each
PA is a CW subcomplex we have that {PA | A ∈ PfV } is a CW cover of PX and so by
Proposition 1.1.2 we are done.
Lemma 1.2.7. Let Γ = (V,E) be a graph and let G be a V -tuple of groups. Then GΓ is
the colimit of the diagram which
• has objects the groups Gv for each v ∈ V and Ge := Gv1×Gv2 for each e = {v1, v2} ∈
E, and
• has morphisms the inclusions
φ : Gv ↪→ Ge (1.2.3.6)
for each v ∈ e = {v, w}, where φ(g) = (g, e).
Proof. We proceed by induction on the number of edges e. When e = 0, the diagram is
a set of objects {Gv | v ∈ V } and so the colimit is the free product of the vertex groups
as desired.
Now suppose the lemma holds for any graph with e = m− 1 edges and let Γ = (V,E)
be a graph with m edges. Choose an edge {v, w} ∈ E and let Γ′ be the graph with edge
set E−{v, w}. Since Γ′ has m−1 edges we know by the assumption that GΓ′ ∼= colimD′
for the associated diagram of groups D′. The diagram D associated to Γ consists of D′
along with the object Gv × Gw and morphisms Gv → Gv × Gw ← Gw. So the colimit of
D is the colimit of
colimD′ Gv ×Gw
Gv
OO 88pppppppppppp
Gw.
OOffNNNNNNNNNNN
(1.2.3.7)
But the colimit of this is
GΓ′
〈[g, h] | g ∈ Gv, h ∈ Gw〉 (1.2.3.8)
16 1. PRELIMINARIES
which is isomorphic to GΓ. Thus by induction we are done.
The proof of the theorem then follows
Proof of Theorem 1.2.5. Lemma 1.2.6 realises PX as a colimit of a diagram DP of pointed
spaces. Meanwhile Lemma 1.2.7 realises GΓX as the colimit of a diagram D
G
Γ . The
fundamental group functor preserves colimits, so the fundamental group of PX is the
colimit of pi1(D
P) which we will denote DG. So we have reduced the Theorem to proving
that the colimits of DGΓ and of D
G are isomorphic.
Comparing the two diagrams we find that DGΓ is the full subdiagram of D
G consisting
of the binary products and the vertex groups themselves. For an object not in the subdi-
agram, GA =
∏
v∈AGv for A ⊆ [n] with |A| ≥ 3, there are natural maps Gv ×Gw → GA
for each pair {v, w} ⊂ A. And inside the group GΓ these groups Gv × Gw generate a
subgroup isomorphic to GA. Hence there is a map from GA to the colimit of D
G
Γ for
every subset A. And so a unique extension from the colimit of DG to that of DGΓ . This
implies that the two colimits are isomorphic.
1.2.4 Flag complexes and classifying spaces
We have seen that the fundamental group of a simplicial complex product of pointed
spaces is equal to the graph product of the fundamental groups of the pointed spaces,
where the graph is obtained from the simplicial complex by taking the 1-skeleton. A
related problem is to go the other way. Given a graph product of groups, can one find
a simplicial complex and pointed spaces such that the simplicial complex product is a
classifying space for the graph product? The answer is that we can.
Recall that a flag complex is a simplicial complex which may be constructed from its
1-skeleton graph, in fact it is the maximal simplicial complex for a fixed graph.
Theorem 1.2.8. Let Γ = (V,E) be a finite graph, let X = FlΓ be its corresponding flag
complex and let P be a V -tuple of classifying spaces for G a V -tuple of groups. Then the
simplicial complex product PX is a classifying space for GΓ.
This will be proved shortly. First we introduce some terminology.
Links and stars
Definition 1.2.9. Let X = (V, S) be a simplicial complex and v be a vertex in V . The
star at v is defined to be
{A ∈ S | A ∪ {v} ∈ S}. (1.2.4.1)
Then the link at v is defined to be the simplicial complex
{A ∈ S | v /∈ A and A ∪ {v} ∈ S}. (1.2.4.2)
1.2. GRAPH PRODUCTS AND RIGHT-ANGLED ARTIN GROUPS 17
The cone of X is defined to be the simplicial complex C(X) = (V ∪ {0}, S ′) where
S ′ = S ∪ {A ∪ {0} | A ∈ S}. (1.2.4.3)
It is immediate from the definition that st(v) is the cone of lk(v) where v takes the
role of the point 0.
Example 1.2.4.1. Consider the unique simplicial complex with 4 vertices, 4 edges and
a single triangle (or 2-simplex). We draw below the simplicial complex, followed by the
star at v, followed by the link at v.
v v (1.2.4.4)
The next example is the same but with the triangle missing. We again show the simplicial
complex, followed by the star at v, followed by the link at v.
v v (1.2.4.5)
Lemma 1.2.10. Let X be a flag complex and v be a vertex in V . Then both the link and
star at v are full subcomplexes of X and hence themselves flag complexes.
Proof. Let A be a simplex in X such that each w ∈ A is contained in lk(v). To prove
that lk(v) is a full subcomplex of X we need only prove that A is also a simplex of lk(v).
By the definition of the link at v we have that for each w ∈ lk(v) the pair {v, w} is
contained in X. But then A ∪ {v} spans a complete graph in the 1-skeleton of X, hence
by the flag condition A ∪ {v} is a simplex of X. Therefore A is contained in the link at
v, which is what we needed to prove.
The argument for st(v) is identical. The fact that a full subcomplex of a flag complex
is itself a flag complex is immediate from the definition.
Proof of Theorem 1.2.8. We proceed by induction on the number of vertices n of Γ. When
n = 1 the theorem is trivially true. Now assume that it is true for m < n and let
Γ be a graph with n vertices. Let v be a vertex of Γ. Define Xv to be the simplicial
complex consisting of the simplices of X which do not contain v. This is the flag complex
corresponding to the graph Γv obtained by removing the vertex v and all adjacent edges
{v, w}. Of course Γv has n − 1 vertices and so by the assumption the space PXv is a
18 1. PRELIMINARIES
classifying space for GΓv. Using Lemma 1.2.10 we may also apply the assumption to the
link at v denoted lk(v) which also does not contain v so has n − 1 or fewer vertices: we
get that P lkx is a classifying space for the group GΓlk(v) where Γlk(v) is the subgraph of
Γ spanned by vertices which are joined by an edge to v.
The star at v denoted st(v) may have n vertices so we can not apply the induction
hypothesis. We may however note that since the star is the cone of the link we have that
P st(v) ∼= P lk(v)× Pv. (1.2.4.6)
Hence P st(v) is also a classifying space of G st(v).
Both P st(v) and PXv are CW subspaces of PX and in fact they cover PX and have
intersection P lk(v), so we may apply Proposition 1.1.2 to get that PX is the colimit of
the diagram
PXv P lk(v)oo // P st(v) (1.2.4.7)
and so on fundamental groups we have that GΓ is the amalgamation of
GΓv GΓlk(v)oo //GΓlk(v) ×Gv, (1.2.4.8)
using (1.2.4.6) to decompose pi1(P st(v)) as a direct product. The right-hand arrow is
clearly injective, to see that the left-hand arrow is also injective we may just note that
there is a one-sided inverse given by sending any vertex group element corresponding
to a vertex not in Γlk(v) to the identity. We have now satisfied all of the conditions of
Theorem 1.1.1 and so PX is a classifying space for GΓ and we are done by induction.
Now that we can construct classifying spaces for graph products of groups we can
compute invariants such as the homology of the groups.
Corollary 1.2.11. Let Γ be a finite graph and G be a V -tuple of groups. Then for any
commutative ring R
H∗(GΓ, R) = R⊕
⊕
K
H∗
(⊗
v∈K
Ĉ∗(Gv, R)
)
, (1.2.4.9)
where K ranges over the complete subgraphs of Γ and Ĉ∗(Gv, R) is an R-projective complex
computing the reduced homology of Gv.
The (co)homology of the classifying space PFlΓ can be calculated using Theorem 2.3.2.
Since the proof for simplicial complexes does not significantly simplify from the general
case of diagonal complexes we will not rewrite it here.
1.2. GRAPH PRODUCTS AND RIGHT-ANGLED ARTIN GROUPS 19
1.2.5 Summary
The following diagram summarises this introductory section. This is analagous to our
approach to diagonal complexes in the next chapter, so it will be very helpful if the reader
understands the arguments presented in Section 1.2.
Subspace of
∏
v Yv
Covering by
product spaces
Colimit diagram
xxqqq
qqq
qqq
qqq
qq
X a flag complex
))SSS
SSSS
SSSS
SSS
pi1 H∗ and H∗
decomposition into
smaller pieces
Yv aspherical
Proof of asphericity.
(1.2.5.1)
20 1. PRELIMINARIES
Chapter 2
Diagonal complexes
In the introductory chapter we studied the graph product of groups and set up a theory
to compute their homology. We now proceed to a more general class of products of
groups. Whereas graphs describe RAAGs, diagonal complexes describe ‘diagonal right-
angled Artin groups’, or DRAAGs. The draw of this new class of groups is present for a
number of reasons:
1. groups that we already know well are included in their number,
2. most of the structures used to study RAAGs carry over in a simple manner,
3. the potential structures of the groups are more intricate than the RAAGs and the
methods used, such as CAT(0) geometry, do not carry over to the DRAAGs.
I hope that the DRAAGs provide fellow group theorists with interesting new examples
to explore. In particular we uncover examples of groups with simple, naturally defined
classifying spaces built out of tori, but which do not appear to have naturally defined
non-positively curved metrics.
However the original motivation for defining diagonal complexes came from a partic-
ular example. We wished to provide a solid combinatorial framework to
1. describe moduli spaces of cactus products (defined in Chapter 3) and then to
2. prove the asphericity of the moduli spaces and to compute their homology.
It soon became clear that the notion arrived at was a good generalization of the graph
product of groups and that the theory encompassed a wide variety of groups sharing some
properties of graph products, but not all.
Since many interesting spaces (including the moduli spaces) can be embedded into
product spaces as unions of diagonal maps we will first describe the poset of partial
partitions under partial coarsening. In Theorem 2.1.2 we show that every union of diagonal
maps in a product space is uniquely described as a subposet closed from beneath. However
21
22 2. DIAGONAL COMPLEXES
this notion is too general for our purposes; the fundamental groups are tricky to describe
and their homological properties hard to compute.
The choice of definition of diagonal complexes restricts the spaces we are able to
describe, on the other hand the spaces we get are more pleasant to work with. Also the
posets involved are subsets of a power set, rather than subsets of the set of all partial
partitions. As such we need only work with ‘relatively small’ combinatorial structures.
2.1 Definitions
2.1.1 Diagonal maps
Let Y be a set, topological space or group. In each case there is a diagonal map1
∆Y : Y → Y × Y
y 7→ (y, y)
(2.1.1.1)
from Y to the direct product Y × Y . We will denote by ∆n−1Y the map from Y to the
n-fold direct product Y ×n,
∆n−1Y (y) = (y, . . . , y). (2.1.1.2)
In particular when n = 1 then ∆0Y is the identity on Y .
Now suppose that Y is pointed, that is there is a chosen map
p : {1} 7→ Y. (2.1.1.3)
In the case where Y is a group there is one choice of map which is the unique map from
the trivial group.
We are interested in the natural maps which can be built between Y ×k and Y ×n from
the diagonal map ∆Y and the point map p. Since both generating maps increase the
number of copies of Y by 1 we require that k ≤ n. Such maps can be written as a product
∆n1−1Y ×∆n2−1Y × . . .×∆nk−1Y × pm, where n1 + . . .+ nk +m = n, (2.1.1.4)
composed with a permutation σ ∈ Sn which permutes the factors of Y n. These maps
can be denoted by an ordered partition (U1, . . . , Uk, B) of [n]. To see this apply the
permutation σ ∈ Sn to the partition
({1, . . . , n1}, {n1+1, . . . , n1+n2}, . . . , {n1+. . .+nk−1+1, . . . , n1+. . .+nk},
{n−m+1, . . . , n}) (2.1.1.5)
1In fact in any category with binary products there exist such diagonal maps.
2.1. DEFINITIONS 23
Conversely given an ordered partition (U1, . . . , Uk, B) choose a permutation σ ∈ Sn which
sends it to the form (2.1.1.5) where ni =|Ui |. The corresponding map is then (2.1.1.4)
followed by σ−1 permuting the factors of Y n. Since ∆Y is symmetric the choice of σ does
not affect the given map. We will denote the map by
D
(U1,...,Uk)
Y (2.1.1.6)
In practice we use partial partitions {U1, . . . , Uk} of [n] because B can be recovered
by taking the complement of the union of the Ui’s in [n]. We choose to forget the or-
dering because although the ordered (U1, . . . , Uk) determines the map Y
×k → Y ×n we
are interested in the image of the maps which depends only on the unordered partial
partitions. We call the image of such a map a diagonal subspace (or diagonal subset, or
diagonal subgroup depending on what type of object we are interested in). The number
k is referred to as the rank of the diagonal subset.
We may illustrate this with an example; the following is a list of all six partial partitions
of [3] = {1, 2, 3} consisting of two subsets, along with a choice of image of (x, y) in Y ×3.
1 | 2 (x, y, 1)
1 | 3 (x, 1, y)
2 | 3 (1, x, y)
12 | 3 (x, x, y)
13 | 2 (x, y, x)
23 | 1 (y, x, x)
(2.1.1.7)
So these six partial partitions correspond to the six copies of Y ×2 inside Y ×3 naturally
given by diagonal and point maps or in other words, the diagonal subspaces of rank
2. These are pictured for Y = I the unit interval below. The first three ‘orthogonal’
subspaces are pictured in the first cube, the remaining three are pictured in the other
cubes in no particular order.
(2.1.1.8)
2.1.2 The poset of partial partitions
We saw above that the diagonal subspaces of a product Y n were given by partial partitions
of [n]. The aim is to construct a theory to study these subspaces and their unions. The
24 2. DIAGONAL COMPLEXES
union of two diagonal subspaces is not a diagonal subspace unless one is contained in
the other, but their intersection is. We devote this section to understanding this meet-
semilattice2 and it turns out that we need not look further than the set of partial partitions
and a partial order on this set.
Denote the set of all subsets of a set X by PX and the set of all finite subsets by
PfX. Recall that a partition of X is a set of subsets {Ui ⊆ X | i ∈ I} which are pairwise
disjoint and whose union is X. A partial partition of X is a partition of a subset of X.
Alternatively a partial partition is a set {Ui} of pairwise disjoint subsets. In this thesis
all partial partitions will be partitions of finite subsets of X. Denote by PPfX the set of
partial partitions whose union is a finite set. We will refer to a subset {Uj | j ∈ J} of a
partial partition {Ui | i ∈ I} for J ⊆ I as a subpartition of {Ui}. Let
D
{Ui}
Y : Y
I → Y X (2.1.2.1)
represent the diagonal subspace, then precomposing this with the natural inclusion
idJY × pI−JY : Y J → Y I (2.1.2.2)
one gets a representative of the diagonal subspace corresponding to {Uj | j ∈ J}. So being
a subpartition implies the inclusion of diagonal subspaces, however the converse does not
hold so we require a finer ordering on PPfX to fully represent the diagonal subspaces.
Suppose that {Ui} and {Vj} are two elements of PPfX. Then we say that {Ui} is a
coarsening of {Vj} (or that {Vj} is a refinement of {Ui}) if the unions are equal and if for
each Vj there is a Ui containing Vj. If {Ui} is a coarsening of a subpartition of {Vj} then
we say that {Ui} is a partial coarsening of {Vj} and write {Ui} ≤pc {Vj}.
Lemma 2.1.1. The pair (PPfX,≤pc) forms a poset.
Proof. The reflexivity condition is clear, {Ui} is a coarsening of {Ui}. Now for antisym-
metry, suppose that both {Ui} ≤pc {Vj} and {Vj} ≤pc {Ui}, then both their unions are
contained in one another and so must be equal, hence they are coarsenings of each other.
Now take some Ui ∈ {Ui}, since Vj is a coarsening of Ui there exists a Vj ∈ {Vj} containing
Ui, but {Ui} is a coarsening of {Vj} so there exists a Uk containing Vj and hence Ui. But
the subsets in {Ui} are pairwise disjoint, so Ui = Uk and since Vj is sandwiched between
them Vj = Ui. This is true for each Ui so {Ui} = {Vj}.
Finally transitivity, suppose that {Ui} ≤pc {Vj} ≤pc {Wk} and pick a Wk ∈ {Wk}.
Either Wk does not intersect {Vj} in which case it does not intersect {Ui} or there is some
Vj containing it, suppose the latter. Now either Vj does not intersect {Ui} in which case
Wk does not intersect {Ui} or there is some Ui containing Vj and hence also Wk. We are
2a meet-semilattice is a poset which has a meet (a greatest lower bound) for any non-empty finite
subset.
2.1. DEFINITIONS 25
left with the possibility either that Wk does not intersect {Ui} or that there is some Ui
containing Wk, hence {Ui} ≤pc {Wk}.
We will now examine the diagonal subspaces again. Let {U1, . . . , Uk} be a partial
partition of [n] then we have a diagonal subspace the image of the map
D
{Ui}
Y : Y
k → Y n. (2.1.2.3)
Recall that this may be given by
D
{Ui}
Y (y1, . . . , yk) = (x1, . . . , xn), (2.1.2.4)
where
xi =
yj if i ∈ Uj,∗ if i /∈ Uj for each j = 1, . . . , k. . (2.1.2.5)
The image may be characterised by{
(yi)i∈[n] | yi = yj for i, j ∈ Ui and yl = ∗ for l /∈
⋃
i
Ui
}
. (2.1.2.6)
Theorem 2.1.2. Let Y be a pointed space containing some point y 6= ∗. Then the poset
(DS,⊆) consisting of the diagonal subspaces of Y ×n ordered by inclusion is isomorphic to
the poset (PPf [n] ,≤pc). Furthermore the intersection of two diagonal subspaces is itself
a diagonal subspace and so meets are defined in (PPf [n] ,≤pc).
Proof. A diagonal subspace is the image of a direct sum of diagonal and point maps
which as discussed in Section 2.1.1 is given by a partial partition. Showing that two
partial partitions give different diagonal subspaces will tell us that the sets PPf [n] and
DS are equal. We then show that these two sets are also isomorphic as posets. So let {Ui}
and {Vj} be distinct partial partitions. If the unions are distinct then we may assume
that
⋃
i Ui 6⊆
⋃
j Vj, then the element (yx)x∈X with
yx =
y if x ∈
⋃
i Ui,
∗ otherwise
(2.1.2.7)
is in ImD
{Ui}
Y but not in ImD
{Vj}
Y , which can be seen by examining (2.1.2.6). Suppose now
that the unions are equal. Since {Ui} and {Vj} are distinct we may find Ui and Vj such
that Ui∩Vj 6= ∅ and Ui 6= Vj. Suppose that neither is contained in the other then (yx)x∈X
with
yx =
y if x ∈ Ui,∗ otherwise (2.1.2.8)
26 2. DIAGONAL COMPLEXES
is in ImD
{Ui}
Y but not in ImD
{Vj}
Y . We are left with the case that one is contained in
the other, so assume that Ui ⊂ Vj, then the same element (2.1.2.8) serves to distinguish
ImD
{Ui}
Y and ImD
{Vj}
Y .
We have now shown that the set of diagonal subspaces is equal to the set of partial par-
titions. If {Ui} ≤pc {Vj} then by (2.1.2.6) we have the containment ImD{Ui}Y ⊆ ImD{Vj}Y .
Now for the converse, suppose that {Ui}, {Vj} ∈ PPf [n] and ImD{Ui}Y ⊆ ImD{Vj}Y , we need
to show that {Ui} ≤pc {Vj}.
If the union
⋃
i Ui was not contained in
⋃
j Vj then the element defined in (2.1.2.7)
would contradict the inclusion. Finally for any Ui consider the element of ImD
{Ui}
Y defined
by (2.1.2.8). Since this element is in ImD
{Vj}
Y , the set Ui must be a union of some of the
{Vj}. We have shown the equality of the two posets.
For the second part of the theorem we will describe the intersection of two subspaces
ImD
{Ui}
Y and ImD
{Vj}
Y and then remark that it is given by the meet of {Ui} and {Vj} in
(PPf [n] ,≤pc). Let (yx)x∈X ∈ ImD{Ui}Y ∩ ImD{Vj}Y , then if either a, b ∈ Ui or a, b ∈ Vj, then
ya = yb. So define an equivalence relation on [n] ∪ {0} as follows, write a ∼ b if either
a, b ∈ Ui or a, b ∈ Vj for some Ui or Vj. Also write a ∼ 0 if either a /∈
⋃
i Ui or a /∈
⋃
j Vj.
Extend ∼ to an equivalence relation by transitivity. If a ∼ b in ([n]∪{0},∼) then ya = yb
in (yx)x∈X .
Define {Wk} to be the equivalence classes of ∼ not containing 0. Then (yx)x∈X is in
ImD
{Wk}
Y , also any element of ImD
{Wk}
Y is certainly in ImD
{Ui}
Y and ImD
{Vj}
Y . The partial
partition {Wk} is the meet of {Ui} and {Vj} in (PPf [n] ,≤pc) so we have
ImD
{Ui}
Y ∩ ImD{Vj}Y = ImD{Ui}∧{Vj}Y . (2.1.2.9)
We set out to describe the spaces which can be built as unions of diagonal subspaces of a
product space. Using the previous theorem we have our answer, such spaces are described
by the subposets of (PPf [n] ,≤pc) which are closed in the sense that if {Ui} ≤pc {Vj} and
{Vj} is in the poset, then {Ui} is also in the poset.
2.1.3 Diagonal complexes
Theorem 2.1.2 above described any space built as the union of diagonal subspaces. We now
turn our attention to a smaller class of such unions, those given by diagonal complexes.
The main body of this chapter is devoted to describing their homotopical invariants.
Let X be a set and write X+ for
{
x+ = {x} | x ∈ X} the set of singleton subsets
of X.
Definition 2.1.3. A diagonal complex on X consists of a pair (Γ, γ), with Γ ⊆ PfX and
γ : Γ→ PfΓ such that
1. X+ ⊆ Γ,
2.1. DEFINITIONS 27
2. for each U ∈ Γ the set γ(U) is a partition of U . So the image of γ is contained in
PPfX. Furthermore if U is not a singleton then γ(U) is a proper partition. That
is, if U ∈ Γ−X+ then |γ(U)|> 1,
3. (simplicial condition) for U ∈ Γ we write γ(U) = {U1, . . . , Uk}. For each A ⊆ [k]
we require that UA ∈ Γ, where
UA :=
⋃
i∈A
Ui. (2.1.3.1)
We also require that γ(UA) is either {Ui | i ∈ A} or a refinement of {Ui | i ∈ A}. We
call the UA the faces of U .
The dimension of U ∈ Γ is defined to be |γ(U)|. Note that the dimension of a face of U
may be greater than the dimension of U .
Consider the poset on Γ transitively generated by V ≤ U if V is a face of U . In this
poset if V ≤ U we say that V is a descendant of U . For example a face of a face of U is
a descendant, but not necessarily a face of U . If a diagonal complex (Γ, γ) satisfies
4. the descendance order agrees with the ordering by inclusion.
then we say that (Γ, γ) is proper.
Remark 2.1.4. Condition 1. in the above definition is not necessary for many of the
results in this thesis. However asking for it gives the diagonal complex product of groups
a minimal set of generators.
Condition 2. says that γ has image in the intersection of PfΓ and PPfX, in fact it
implies that γ is a setwise section of the map of posets (PPfX,≤pc)→ (PfX,⊆) given by
taking the union of a partial partition.
A diagonal complex which is proper is a convenient object to work with because by
the proposition below the inclusion ordering on Γ determines the map γ. With this in
mind we will sometimes just write Γ for (Γ, γ) when it is proper.
Proposition 2.1.5. Let (Γ, γ) be a diagonal complex on a set X. Then (Γ, γ) is proper
if and only if for each U ∈ Γ
γ(U) = {U −M1, . . . , U −Mk}, (2.1.3.2)
where M1, . . . ,Mk are the set of maximal subsets under U in the inclusion order on Γ.
Proof. Let U ∈ Γ, then γ(U) = {U1, . . . , Uk} is a partition of U . Recall that the unions
of the Ui are called the faces and so the maximal faces are the unions⋃
i 6=j
Ui (2.1.3.3)
28 2. DIAGONAL COMPLEXES
for each j. But this is just U − Uj and by the definition of the descendance order these
are maximal under U in the descendance order.
Now suppose that (Γ, γ) is proper, then the inclusion order agrees with the descendance
order and the sets U − Uj are maximal subsets of U .
Conversely, if (2.1.3.2) holds then U − Uj are both maximal in the descendents order
and as subsets of U for every U ∈ Γ. Such a property of finite posets implies that the two
orderings are equal.
2.1.4 The geometric realisation
A diagonal complex (Γ, γ) over a set X may be encoded as a diagram called the geometric
realisation consisting of a subset of the simplex ∆X with vertex set X. In fact the
realisation is a sub-simplicial complex of the barycentric subdivision of ∆X . Recall that
the barycentric subdivision of a simplex with vertex set X has vertex set PfX, the set
of non-empty finite subsets of X. Then a simplex in the subdivision is a set of subsets
which form a total order under the partial order by inclusion:
A1 ⊂ A2 ⊂ . . . ⊂ Ak. (2.1.4.1)
Now suppose that {Ui} is a partial partition of X. Then each Ui corresponds to a vertex
of the barycentric subdivision. If ∆X is embedded in RX as the convex hull of the vectors
{ex | x ∈ X} then the vertex corresponding to Ui is given by
eUi =
1
|Ui|
∑
x∈Ui
ex (2.1.4.2)
We may associate a subspace to {Ui} by taking the convex hull of the eUi , call this simplex
S({Ui}). This is infact spanned by simplices in the barycentric subdivision of ∆X , the
vertices are all possible unions of the Ui’s.
Definition 2.1.6. To (Γ, γ) associate a subset of ∆X :
|(Γ, γ)| =
⋃
U∈Γ
S(γ(U)). (2.1.4.3)
We call this the geometric realisation of (Γ, γ).
By the discussion above this realisation can also be viewed as a realisation of some
sub-simplicial complex of the barycentric subdivision of ∆X .
2.1. DEFINITIONS 29
2.1.5 Examples
The following list of examples should help to build up some intuition for the kind of
structures we can build.
Example 2.1.5.1. Let A = (X,S) be a simplicial complex with vertex set X. Let Γ = S
and define γ(U) = U+ =
{
x+ = {x} | x ∈ U}. Then (Γ, γ) is a proper diagonal complex.
It’s geometric realisation is isomorphic to the geometric realisation of A.
Example 2.1.5.2. Let X = [3], we define Γ to be the set
{{1, 2, 3} = X, {1, 2}}∪X+ and
γ is defined by γ({1, 2, 3}) = {{1, 2}, 3+} and γ({1, 2}) = {1+, 2+}. We may represent
(Γ, γ) by the geometric realisation:
3
1 2
(2.1.5.1)
The descendance order agrees with the inclusion order and so Γ is proper.
Example 2.1.5.3. An example of a diagonal complex which is not proper is given by
adding the element {1, 3} onto the previous example. The picture we now get is
3
1 2
(2.1.5.2)
Note that {1, 3} is a subset of {1, 2, 3} but it is not a descendant and so this diagonal
complex is not proper.
Example 2.1.5.4. The set X can be infinite, for example let X = {1, 2, . . .} be the
strictly positive natural numbers. Define Γ to be
{{1, . . . , n} | n = 1, 2, . . .} ∪X+ (2.1.5.3)
and γ on the non-singleton sets by
γ
({1, . . . , n}) = {{1, . . . , n− 1}, {n}}. (2.1.5.4)
So the diagonal complex (Γ, γ) consists of subsets of unbounded cardinality, however
the dimension of the subsets is bounded by two. It can be represented by the following
30 2. DIAGONAL COMPLEXES
diagram
1
2
3
4
5 7
6 8
(2.1.5.5)
2.1.6 Morphisms of diagonal complexes
The morphisms between diagonal complexes are important because with a good definition
of a morphism the diagonal complex product (to be defined in the next section) will be
functorial and so morphisms of diagonal complexes will induce maps between diagonal
complex products.
So let (Γ1, γ1) and (Γ2, γ2) be diagonal complexes over sets X1 and X2 respectively.
Then a morphism
f : (Γ1, γ1)→ (Γ2, γ2) (2.1.6.1)
is given by a map of sets also denoted f
f : X1 → Γ2 (2.1.6.2)
which may be extended to a function denoted F
F : Γ1 → Γ2 (2.1.6.3)
by taking unions. This must make the following diagram commute
Γ1
F //
γ1
Γ2
γ2
PfΓ1
PfF // PfΓ2.
(2.1.6.4)
The composition of morphisms f : (Γ1, γ1) → (Γ2, γ2) and g : (Γ2, γ2) → (Γ3, γ3) is given
by the composition
X1
g◦f
66
f // Γ2
G // Γ3 (2.1.6.5)
Lemma 2.1.7. The map from Γ1 to Γ3 induced from g ◦ f is precisely G ◦ F and this
2.1. DEFINITIONS 31
map satisfies the diagram (2.1.6.4).
It is now a simple task to check that we have defined a category of diagonal complexes.
Now suppose that f : (Γ1, γ1) → (Γ2, γ2) may be given by an inclusion of vertex sets
like so
X1 ↪→ X2 ∼= X+2 ↪→ Γ2. (2.1.6.6)
For such f ’s we say that (Γ1, γ1) is a subcomplex of (Γ2, γ2). Suppose now that Y ⊆ X is a
subset of X and that (Γ, γ) is defined over X. Let ΓY = Γ∩PfY , then we call (ΓY , γ |ΓY )
the full diagonal subcomplex on Y .
Example 2.1.6.1. Let (Γ, γ) be the diagonal complex defined in Example 2.1.5.2 with
diagram
3
1 2
(2.1.6.7)
Then up to isomorphism there are two injective maps from the line ΓL = {{s}, {t}, {s, t}}.
One sends s to {1} and t to {2} and realises ΓL as the full diagonal subcomplex on {1, 2}.
The other sends s to {3} and t to {1, 2}.
Levels of a diagonal complex
Let (Γ, γ) be a diagonal complex on a set X. We will define the level lev : Γ → N
inductively as follows:
• the level is zero if U ∈ X+, otherwise
• the level is defined to be the maximum of the levels of the maximal subfaces of U ,
plus one:
lev(U) = sup {lev(U − V ) | V ∈ γ(U)}+ 1. (2.1.6.8)
The level is well-defined because for U ∈ Γ−X+, the cardinalities of the elements of γ(U)
are strictly less than the cardinality of U . The level will be used in inductive arguments.
For n ∈ N we define (Γn, γn) to be the diagonal subcomplex given by elements of level
n or below. This defines a filtration of (Γ, γ):
(X+ = Γ0, γ0) ≤ (Γ1, γ1) ≤ (Γ2, γ2) ≤ . . . . (2.1.6.9)
The union is the whole of (Γ, γ).
Remark 2.1.8. There is also a coarse level levc : Γ → N which is similarly defined with
levc(X
+) = {0} and
levc(U) = sup {levc(V ) | V ∈ γ(U)}+ 1. (2.1.6.10)
32 2. DIAGONAL COMPLEXES
This defines the coarse filtration (Γci , γ
c
i ). It is worth noting that the zeroth term of both
filtrations consists of just the points of X, whilst in the coarse case the first term comes
from a simplicial complex Γc1 and in the regular case the first term Γ1 is the 1-skeleton of
Γc1. The coarse level will not be used in the sequel although it would be sufficient in some
arguments.
Example 2.1.6.2. The diagonal complex (Γ, γ) of Example 2.1.5.4 defined over X =
{1, 2, . . .} has the levelwise filtration given by
Γn = X
+ ∪ {{1, . . . ,m} | m = 2, . . . , n+ 1} (2.1.6.11)
2.1.7 Products
The very reason for defining diagonal complexes is to study the diagonal complex prod-
ucts they index and which we define in this section. We will begin by constructing the
products for pointed spaces and then by considering this case expand the definition to
other categories.
Labelled diagonal complexes
Before defining products of spaces we must consider the spaces we are taking the product
over. For this we require a system for labelling the factor spaces.
Definition 2.1.9. Let (Γ, γ) be a diagonal complex defined over a set X and let l : X → Z
be a map of sets with codomain Z. Then (Γ, γ, l) is a Z-labelled diagonal complex if
• for each U ∈ Γ and each V ∈ γ(U), the map l is constant on V .
We call l the labelling of (Γ, γ).
Example 2.1.7.1. 1. Let Z = {1} then any diagonal complex is Z-labelled with the
constant map X → {1}.
2. Let Γ be the proper diagonal complex defined by a simplicial complex S• over a set
X. See Example 2.1.5.1 for details. Then any labelling l : X → Z for any Z makes
(Γ, γ, l) into a Z-labelled diagonal complex.
3. Let Γ be the proper diagonal complex given in Example 2.1.5.2. Let Z = {•, ◦} and
define l to be
l(1) = •, l(2) = • and l(3) = ◦. (2.1.7.1)
Then (Γ, γ, l) is a {•, ◦}-labelled complex.
2.1. DEFINITIONS 33
In fact each diagonal complex (Γ, γ) defined over X has a universal labelling l0 : X →
Z0 with the property that if l : X → Z is another valid labelling then there exists a
factorisation
X
l0 //
l
66Z0 // Z. (2.1.7.2)
It may be defined as follows, consider the subsets V ⊆ X which occur as V ∈ γ(U)
for some U ∈ Γ. Now define an equivalence relation on X by saying that x and y are
equivalent if they both lie in such a V and extending by transitivity. Now let Z0 be the
set of equivalence classes with l0 : X → Z0 the natural map. It is not hard to see that
this enjoys the above universal property.
Let l : X → Z be a map of sets and Y = (Yi | i ∈ Z) a Z-tuple of pointed spaces. Let
{Ui | i = 1, . . . , k} be a partial partition of X such that l is constant on each Ui, we write
the value as l(Ui). There is now a diagonal map
D
{Ui}
Y :
k∏
i=1
Yl(Ui) →
∏
i∈X
Yl(i). (2.1.7.3)
Suppose that we are given two partial partitions valid with respect to a labelling. The
next lemma says that the meet of these partial partitions is also valid with respect to the
labelling. This should come as no surprise because in light of Theorem 2.1.2 this is the
intersection of the two diagonal subspaces of the form (2.1.7.3).
Lemma 2.1.10. Let X be a set with a labelling l : X → Z. Suppose that {Ui} and {Vj}
are two partial partitions with the property that l is constant on any W in {Ui} or {Vj}.
Then the meet {Wk} = {Ui} ∧ {Vj} in the poset (PPfX,≤pc) also has the property that l
is constant for any W ∈ {Wk}.
Proof. Recall that {Wk}may be constructed by defining an equivalence relation of X∪{0}
with generators given by the sets Ui and Vj and also the complements of the partial
partitions inside X ∪ {0}. Since l is constant on the generating sets of the form Ui and
Vj, l must be constant on the equivalence classes which do not include 0. But these
equivalence classes are precisely the elements of {Wk}.
Preliminary construction
Let Y = (Yj)j∈Z be a Z-tuple of pointed spaces and let (Γ, γ, l) be a Z-labelled diagonal
complex on X, then define YX to be
{
(yi)i∈X | yi 6= ∗l(i) for finitely many i ∈ X
} ⊆ ∏
i∈X
Yl(i). (2.1.7.4)
34 2. DIAGONAL COMPLEXES
Recall that diagonal subspaces were defined by partial partitions of X. For each U ∈ Γ
we are given a partial partition γ(U) of X and this defines a map
D
γ(U)
Y : Y
γ(U) :=
∏
U ′∈γ(U)
Yl(U ′) → YX . (2.1.7.5)
The space Y′(Γ, γ) is defined as ⋃
U∈Γ
ImD
γ(U)
Y . (2.1.7.6)
This will turn out to be isomorphic to the diagonal complex product of (Yi)i∈Z indexed
by (Γ, γ, l) as proved in Proposition 2.1.12.
Definition as a colimit
Let (Γ, γ, l) be a Z-labelled diagonal complex defined over a set X. Recall that γ has
codomain PPfX. Let PΓ be the subposet of (PPf ,≤pc) containing γ(Γ) and closed under
meets.
Let C be a category with finite products. Then there is an initial object, call it k.
Along with the binary product ⊗ this makes (C, k,⊗) into a symmetric monoidal category.
Furthermore for each object C in C there is a natural diagonal map ∆C : C → C ⊗ C.
Let C = {Ci | i ∈ Z} be a Z-tuple of objects in C.
Now viewing PΓ as a category we define a functor F(C,Γ) as follows:
• An object {Ui} is taken to
⊗
iCl(Ui),
• the image of a morphism {Ui} ≤pc {Vj} is the product of maps⊗
Ui=qk∈IVk
∆
|I|−1
Cl(Ui)
⊗
⊗
Vk 6⊆∪iUi
pCl(Vk) (2.1.7.7)
consisting of a diagonal map for each Ui and an initial morphism for each Vk not
contained in a Ui.
In the category of pointed spaces or groups the maps are those that take (yi) to (y
′
j),
where if Vj ⊆ Ui for some i then y′j = yi ∈ Yl(Ui) = Yl(Vj), otherwise y′j = ∗ ∈ Yl(Vj).
Definition 2.1.11. If the colimit of F(C,Γ) exists then we call it the diagonal complex
product C(Γ, γ) of C indexed by (Γ, γ).
Recall (2.1.7.6) the definition of Y′(Γ, γ).
Proposition 2.1.12. Let C be the category of pointed spaces and Y = C and (Γ, γ, l) be
as in the definition above. Then the diagonal complex product Y(Γ, γ) is isomorphic to
Y′(Γ, γ).
2.1. DEFINITIONS 35
Proof. The product Y′(Γ, γ) is defined to be the union of the inclusions
iU = D
γ(U)
Y : Y
U → YX . (2.1.7.8)
So in particular the images of iU cover the space Y
′(Γ, γ). For U1, . . . , Uk ∈ Γ the
intersection Im iU1 ∩ . . . ∩ Im iUm is given by ImD{Wk}Y where {Wk} is the meet of the
γ(Uj) for j = 1, . . . ,m.
Hence the Im iU form a cover of Y
′(Γ, γ) and the functor F(Y,Γ) : PΓ → {Pointed Spaces}
is the diagram consisting of the spaces Im iU and their intersections in Y
′(Γ, γ). In the
situation that all the spaces and maps are CW complexes this implies that Y′(Γ, γ) ∼=
colimF(Y,Γ) = Y(Γ, γ) by Proposition 1.1.2, see also [25] Section 4.G.
Example 2.1.7.2. We defined a diagonal complex (Γ, γ) in Example 2.1.5.2 and gave a
labelling in Example 2.1.7.1. The category PΓ is
•
•
•
OO
• // • •oo
(2.1.7.9)
and the corresponding diagram for objects C1 and C2 is given by
C2
p⊗1
C1 ⊗ C2
C1
∆
1⊗p
OO
C1
1⊗p // C1 ⊗ C1 C1.p⊗1oo
(2.1.7.10)
where the morphisms from the initial object are denoted p and the identity morphisms
are denoted 1.
Now suppose that each Ci is a line segment I = [0, 1] with basepoint 0 in the category
of pointed spaces. Then the product IΓ looks like:
(2.1.7.11)
36 2. DIAGONAL COMPLEXES
Another definition of the geometric realisation
There are a number of different ways to define the geometric realisation of a diagonal
complex, originally defined in Section 2.1.4. Here is a method using the product associated
to a diagonal complex. Let (Γ, γ) be a diagonal complex over a set X and let I = [0, 1]
be the unit interval with basepoint 0. Then the geometric realisation is equal to the
intersection
I(Γ, γ) ∩ {(ti)i∈X | finitely many ti are non-zero and ∑
i
ti = 1
}
(2.1.7.12)
of the diagonal product of I indexed by (Γ, γ) seen as a subspace of IX and the full
simplicial complex on X embedded in IX . It may be a helpful exercise to examine the
figure above in Example 2.1.7.2 in order to see that the geometric realisation in this case
is a ‘T’ shape.
In fact I(Γ, γ) is a simplicial cone over the basepoint (0). The geometric realisation is
isomorphic to the base of this cone. In fact this gives the realisation as a subcomplex of
the barycentric subdivision of the full simplicial complex on the set X.
Spherical realisation
Now let R be the real line with basepoint chosen to be 0. For a diagonal complex (Γ, γ)
the diagonal complex product R(Γ, γ) is a subspace of
RX = {(xi)i∈X | finitely many xi are non-zero}. (2.1.7.13)
The finite support condition means that RX may be given the standard Euclidean metric,
i.e. that for two points (xi) and (yi) the following is well-defined√∑
i∈X
(xi − yi)2
.
(2.1.7.14)
Definition 2.1.13. The spherical realisation of a diagonal complex (Γ, γ) is the intersec-
tion
R(Γ, γ) ∩ {(xi) |∑
i
x2i = 1
}
(2.1.7.15)
of the unit sphere and the diagonal complex product R(Γ, γ) embedded in RX .
The spherical resolution is of interest because it occurs as the link of the vertex in the
diagonal complex product of copies of the circle. This means it is of interest in the study
of possible geometries on diagonal complex products.
2.2. PRODUCTS OF GROUPS 37
Levelwise construction of products of pointed spaces
We previously defined the level of a U ∈ Γ and saw that there was a filtration
(X+ = Γ0, γ0) ≤ (Γ1, γ1) ≤ (Γ2, γ2) ≤ . . . . (2.1.7.16)
Our aim now is to describe how the diagonal complex product Y(Γn, γn) of pointed
spaces Y may be built from that of Y(Γn−1, γn−1). Let U ∈ (Γn, γn) be of level n and
write γ(U) = {U1, . . . , Uk}. Then the maximal faces U − Ui are all of level n− 1 or less.
For each proper subset A ( [k] there is an inclusion∏
i∈A
Yl(Ui) ↪→
∏
i∈[k]
Yl(Ui) = Y
U , (2.1.7.17)
we call this subspace YUA and the union of such subspaces we will denote δY
U . This is
the subspace consisting of elements of YU where at least one of the coordinates is equal
to ∗. From the definition of a diagonal complex the set UA =
⋃
i∈A Ui is in (Γn−1, γn−1)
and γ(UA) is a refinement of {Ui}i∈A. Therefore there is a map
YUA → YUA → Y(Γn−1, γn−1), (2.1.7.18)
which realises YUA as the intersection of Y
U and YUA in Y(Γn, γn). Taking the union
of the YUA we get that the intersection of Y
U and Y(Γn−1, γn−1) is δYU . So we have a
diagram
YU ← δYU → Y(Γn−1, γn−1) (2.1.7.19)
whose colimit attaches YU onto Y(Γn−1, γn−1). In this way Y(Γn, γn) is given by attaching
each U of level n.
2.2 Products of groups
2.2.1 As a fundamental group
Suppose that C and D are categories with finite products and that G : C → D is a functor
preserving both finite products and colimits. Now let (Γ, γ, l) be a Z-labelled diagonal
complex and C be a Z-tuple of objects of C. Recall that the diagonal complex C(Γ, γ, l)
is defined if the colimit of a certain functor F(C,Γ) exists. Now since G preserves finite
products we have that
G ◦ F(C,Γ) = F(G(C),Γ). (2.2.1.1)
38 2. DIAGONAL COMPLEXES
Since G also preserves colimits we have further that
G(C(Γ, γ, l)) ∼= G(C)(Γ, γ, l). (2.2.1.2)
Now let C be the category of pointed, connected spaces and D be the category of
groups and G = pi1 be the fundamental group functor, which preserves finite products
and colimits. Let (Γ, γ, l) be a Z-labelled diagonal complex as before and let Y be a
Z-tuple of pointed spaces. Also denote the Z-tuple of groups (pi1(Yi)) by G. Then
Equation (2.2.1.2) becomes
pi1(Y(Γ, γ, l)) ∼= G(Γ, γ, l). (2.2.1.3)
In words this is ‘the fundamental group of a diagonal complex product of pointed spaces is
the diagonal complex product of the fundamental groups of the spaces ’.
2.2.2 An explicit presentation
We will now construct a presentation for a diagonal complex product of groups in terms
of the diagonal complex and of the factor groups themselves. This is summarised as
Theorem 2.2.1. Let (Γ, γ, l) be a Z-labelled diagonal complex and let G = (Gi)i∈Z be a
Z-tuple of groups. Then G(Γ, γ) is generated by elements
gU = (g, U) (2.2.2.1)
where U ∈ Γ is such that l is constant on U and g ∈ Gl(U). These are subject to the
relations
gU .hU = (gh)U for any gU , hU , (2.2.2.2)
eU = e for each U, (2.2.2.3)
[gU , hV ] = e for U, V ∈ γ(W ) and W ∈ Γ, (2.2.2.4)
gU = gU1 . . . gUk where γ(U) = {U1, . . . , Uk}. (2.2.2.5)
These relations suffice to present G(Γ, γ). The element gU is given by
gU = D
γ(U)
G ◦∆dimU−1Gl(U) (g). (2.2.2.6)
Proof. To give a presentation of G(Γ, γ) we look to the levelwise construction. The group
G(Γ, γ) is the colimit of the diagram.
G(Γ0, γ0)→ G(Γ1, γ1)→ G(Γ2, γ2)→ . . . . (2.2.2.7)
2.2. PRODUCTS OF GROUPS 39
So it will suffice to prove the theorem for each level, which we will do by induction. Note
that G(Γ0, γ0) is the free product ∗i∈XGi and that G(Γ1, γ1) is a graph product of groups.
So for the case n = 0 the theorem can easily be seen to be true. As with the levelwise
construction of spaces, we construct each successive group by amalgamations,
GU ← δGU → G(Γn−1, γn−1) (2.2.2.8)
where U is of level n, γ(U) = {U1, . . . , Uk}, the group GU is
∏
i∈[k] Gl(Ui) and δG
U is the
colimit of the diagram consisting of all the groups
∏
i∈AGl(Ui) for A ∈ δ∆k. However in
the category of groups, if the dimension of U is greater than two, then δGU ∼= GU . Only
in the case when the dimension of U is two, when
δGU = Gl(U1) ∗Gl(U2) (2.2.2.9)
and so the diagram looks like
Gl(U1) ×Gl(U2) ← Gl(U1) ∗Gl(U2) → G(Γn−1, γn−1). (2.2.2.10)
does taking the colimit of the diagram have any effect. The effect in question is that
of adding commutation relations. Even though the amalgamation may not change the
group it is still a good time to prove that the relations (2.2.2.2)-(2.2.2.5) hold for U . The
group GU is a copy of
∏
U ′∈γ(U) Gl(U ′) and each of the factors are seen to be one of the
F ({U ′}), which map diagonally into F (γ(U ′)). Hence by (2.2.2.6) each of the factors
consists of the elements gU ′ for g ∈ Gl(U ′), and so relation (2.2.2.4) is seen to hold. Now
assume that l(U) 6= 0. Then gU for each g is given by the diagonal Gl(U) → GU , hence
gU =
∏
U ′∈γ(U) gU ′ giving relation (2.2.2.5). Finally relations (2.2.2.2) and (2.2.2.3) are
also given by the inclusion Gl(U) → GU .
Note that the set {gU | U ∈ X+} generates the group and that the remaining gU are
defined for notational convenience.
It will be helpful to examine the presentation in the case of a simple example.
Example 2.2.2.1. Let (Γ, γ, l) be the ‘T’ shaped labelled diagonal complex from Exam-
ple 2.1.7.1 and let G• = Z/(m) and G◦ = Z/(n) be two cyclic groups. Then G(Γ, γ, l) is
the group generated by
g1, g2, g3 and g12 (2.2.2.11)
with relations given by
gm1 = g
m
2 = g
m
12 = e and g
n
3 = e, (2.2.2.12)
coming from the factor groups, the commutator brackets
[g1, g2] = e and [g12, g3] = e, (2.2.2.13)
40 2. DIAGONAL COMPLEXES
coming from the elements {1, 2} and {1, 2, 3} of Γ respectively and finally
g12 = g1g2. (2.2.2.14)
Remark 2.2.2. In fact the group just described is isomorphic to the graph product given
by the graph
1 2 3 (2.2.2.15)
with G1 = G2 = Z/(m) and G3 = Z/(m). If we let the natural generators of this graph
product be x1, x2 and x3 then we may give the isomorphism by sending x1 to g1, x2 to
g12 = g1.g2 and x3 to g3. This isomorphism may be given whenever G• is abelian.
Unfortunately the higher homotopy group functors pii for i ≥ 2 do not preserve col-
imits and so computing the higher homotopy groups of a diagonal complex product is
significantly harder.
2.3 (Co)homology of products
Fortunately computing the homology and cohomology of a diagonal complex is not very
difficult and there is a simple formula. Once this is derived we will then study the ring
structure of the cohomology and finally will look at what happens when there is an action
of a group of automorphisms on the diagonal complex.
2.3.1 (Co)homology
Let Y be a pointed space and let C∗(Y ) be the chain complex of the space Yi over a ring
R. Then C∗(Yi) splits as R ⊕ Ĉ∗(Yi), where Ĉ∗(Yi) is the reduced chain complex. Recall
that there is a decomposition of the homology of a product of pointed spaces:
Proposition 2.3.1. Let Yi be pointed spaces for i = 1, . . . , n. Then the homology of the
product of the pointed spaces decomposes as
H∗
(
n∏
i=1
Yi
)
= R⊕
⊕
A∈∆n
H∗
(⊗
i∈A
Ĉ∗(Yi)
)
, (2.3.1.1)
where ∆n = Pf [n] is the n-simplex.
Proof. Using the definition of the homology and a Ku¨nneth formula we have
H∗
(
n∏
i=1
Yi
)
∼= H∗
(
C∗
(
n∏
i=1
Yi
))
∼= H∗
(
n⊗
i=1
C∗(Yi)
)
2.3. (CO)HOMOLOGY OF PRODUCTS 41
which uses the fact that C∗(Yi) is a free R-module. By using the decomposition C∗(Yi) ∼=
R⊕ Ĉ∗(Yi) we have
n⊗
i=1
C∗(Yi) ∼=
n⊗
i=1
(R⊕ Ĉ∗(Yi))
∼=
⊕
A⊆[n]
⊗
i∈A
Ĉ∗(Yi).
We are done because H∗ commutes with ⊕ and the R term comes from A = ∅ ⊂ [n].
We get a similar decomposition of the homology of a diagonal complex product.
Theorem 2.3.2. Let (Γ, γ) be a Z-labelled diagonal complex on a set X and Y = (Yi)i∈Z
be pointed spaces. The homology of the diagonal complex product Y(Γ, γ) splits as
R⊕
⊕
U∈Γ
H∗(U), (2.3.1.2)
where H∗(U) is given by
H∗
( ⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′))
)
. (2.3.1.3)
Proof. We proceed by induction on the level. For the case of n = 0, the product is a
wedge product of the spaces Yi and so the theorem holds. Now suppose that (2.3.1.2)
holds for level n − 1, that is for the diagonal complex (Γn−1, γn−1). In Section 2.1.7 the
space Y(Γn, γn) was formed by gluing spaces Y
U onto Y(Γn−1, γn−1) using diagrams of
the form
Y(Γn−1, γn−1)← δYU → YU . (2.3.1.4)
Write γ(U) = {U1, . . . , Uk}. Using Proposition 2.3.1 we are able to write
C∗(δYU)→ C∗(YU) (2.3.1.5)
as the split monomorphism
R⊕
⊕
A∈δ∆k
⊗
i∈A
Ĉ∗(Yl(Ui)) ↪→ R⊕
⊕
A∈∆k
⊗
i∈A
Ĉ∗(Yl(Ui)).
For each Ui there is a map Yl(Ui) → YUi ∼= Y dimUil(Ui) given by the diagonal map, and this
induces the inclusion δYU ↪→ Y(Γn−1, γn−1). On chains the map Yl(Ui) → YUi induces an
injection
C∗(Yl(Ui))→ C∗(YUi) (2.3.1.6)
and so the map C∗(δYU)→ C∗(Y(Γn−1, γn−1)) is injective. Taking the colimit of (2.3.1.4)
42 2. DIAGONAL COMPLEXES
has the effect of adding on a term
k⊗
i=1
Ĉ∗(Yl(Ui)), (2.3.1.7)
which gives the term H∗(U) in (2.3.1.2). Thus we have proved the theorem.
Remark 2.3.3 (cohomological version of Theorem 2.3.2). The cohomological version is
similar, although because taking cochains is contravariant, the colimit is replaced by a
limit and so the direct sum should be replaced by the direct product. In the case that Γ
and X are finite the direct product is isomorphic to the direct sum and so we have that
the cohomology of the diagonal complex product Y(Γ, γ) splits as
R⊕
⊕
U∈Γ
H∗(U), (2.3.1.8)
where H∗(U) is given by
H∗
( ⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′))
)
(2.3.1.9)
and
C∗(Yl(U ′), R) ∼= R⊕ Ĉ∗(Yl(U ′)). (2.3.1.10)
Remark 2.3.4. It is worth noting that the proof of the theorem above offers more than a
calculation of the homology of diagonal complex products, there is also a natural quasi-
isomorphism realising this equivalence
R⊕
⊕
U∈Γ
( ⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′))
)
→ C∗
(
Y(Γ, γ)
)
. (2.3.1.11)
The importance of this will become clear in Theorem 2.3.9. There is also a cohomological
version, where of course the arrow is reversed
R⊕
⊕
U∈Γ
( ⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′))
)
← C∗(Y(Γ, γ)). (2.3.1.12)
Hilbert-Poincare´ Series
Let (Γ, γ, l) be a Z-labelled diagonal complex over a finite set X. To each simplex U ∈ Γ
we assign a monomial in the elements of Z = {z1, . . . , zk}:
m(U) =
∏
U ′∈γ(U)
l(U ′). (2.3.1.13)
2.3. (CO)HOMOLOGY OF PRODUCTS 43
The Hilbert-Poincare´ series of (Γ, γ) in the polynomial ring Z[Z] is
h(Γ,γ)(z1, . . . , zk) =
∑
U∈Γ
m(U). (2.3.1.14)
Example 2.3.1.1. The Hilbert-Poincare´ series of Γ with the labelling by {•, ◦} from
Example 2.1.7.1 is
hΓ(z•, z◦) = 2z• + z◦ + z2• + z•z◦. (2.3.1.15)
Let R[[t]] be the complete N-graded ring generated by the indecomposable modules of
R with the product TorR. We use the variable t to keep track of the grading. For example
if R = Z then the indecomposable modules are of the form Z and Z/(pi). The module Z
is the unit of TorZ and we will denote Z/(pi) by xpi . Then
xpi .xqj =
(1 + t)xpi if p = q and i ≤ j,0 if p 6= q (2.3.1.16)
is used to denote the fact that
TorZk
(
Z/(pi),Z/(qj)
) ∼=
0 if k ≥ 2,
Z/(pi) if p = q, i ≤ j and k = 0, 1,
0 if p 6= q and k = 0, 1.
(2.3.1.17)
Now let yi(t) and y
′
i(t) in R[[t]] be the Hilbert-Poincare´ series of H∗(Yi) and H∗(Yi) re-
spectively. From Theorem 2.3.2 we get the following
Corollary 2.3.5. The Hilbert-Poincare´ series of the homology and the cohomology of the
diagonal complex product Y(Γ, γ) are respectively
1 + h(Γ,γ)(y1 − 1, . . . , yk − 1) and 1 + h(Γ,γ)(y′1 − 1, . . . , y′k − 1).
2.3.2 Structure of the cohomology ring
We may use the inclusion Y(Γ, γ) ↪→ YX to calculate the cup product on cohomology.
Lemma 2.3.6. Let (Γ, γ, l) be a Z-labelled diagonal complex defined over a finite set X.
Then the map H∗(YX , R)→ H∗(Y(Γ, γ), R) is surjective.
Proof. By the remarks following Theorem 2.3.2 the cohomology H∗(Y(Γ, γ), R) decom-
poses as
R⊕
⊕
U∈Γ
H∗
(
Ĉ∗(U)
)
. (2.3.2.1)
44 2. DIAGONAL COMPLEXES
And by a cohomological version of Proposition 2.3.1 the complex H∗(YX , R) decomposes
as
R⊕
⊕
A∈∆X
H∗
(⊗
i∈A
Ĉ∗(Yl(i))
)
. (2.3.2.2)
The inclusion of YU =
∏
U ′∈γ(U) Yl(U ′) into
∏
x∈U Yl(x) by diagonal maps induces a map
H∗
(⊗
x∈U
C∗(Yl(x))
)
→ H∗
( ⊗
U ′∈γ(U)
C∗(Yl(U ′))
)
→ H∗
(
Ĉ∗(U)
)
. (2.3.2.3)
So each summand in (2.3.2.1) is mapped to from a summand in (2.3.2.2). It now only
remains to note that since there is a projection from YX to YU providing a one-sided
inverse to the inclusion, the maps (2.3.2.3) on the individual summands are surjective.
The product can now be calculated by considering the diagram
H∗(Y(Γ, γ), R)⊗H∗(Y(Γ, γ), R)
H∗(YX , R)⊗H∗(YX , R)oooo
H∗(Y(Γ, γ), R) H∗(YX , R)oooo
(2.3.2.4)
Given two cocycles in H∗(Y(Γ, γ), R) to multiply, first lift along the surjective map to
cocycles in H∗(YX , R), multiply them there, then map back to H∗(Y(Γ, γ), R).
2.3.3 Extensions by symmetry groups
Let (Γ, γ) be a diagonal complex on a set X, then by inspecting the definition of a
morphism of diagonal complexes we may see that an automorphism of (Γ, γ) must act
by permuting the set X. Now suppose that l : X → Z is a labelling of (Γ, γ), then an
automorphism of the labelled diagonal complex is a permutation σ of X with l ◦ σ = l.
Let G be a group of automorphisms of a Z-labelled diagonal complex (Γ, γ, l) and let
Y be any Z-tuple of pointed spaces. Then G acts on Y(Γ, γ) and fixes the basepoint and
hence G also acts on
pi1(Y(Γ, γ)) ∼= (pi1(Y))(Γ, γ). (2.3.3.1)
The purpose of this section is to calculate the homology of the homotopy quotient of the
action of G on the spaces Y(Γ, γ). The homotopy quotient was reviewed in Section 1.1.3.
2.3. (CO)HOMOLOGY OF PRODUCTS 45
Homology of the homotopy quotient of a diagonal complex
Proposition 2.3.7. Let G act on the Z-labelled diagonal complex (Γ, γ, l) and let Y be
a Z-tuple of pointed spaces. Then
H∗
(
E ×G Y(Γ, γ), R
) ∼= H∗(G,R)⊕ ⊕
U∈Γ/G
H∗
(
Stab(U), Ĉ∗(U)
)
, (2.3.3.2)
where
Ĉ∗(U) =
⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′)) (2.3.3.3)
with the action inherited from the action of Stab(U) on γ(U).
Proof. The homology of E ×G Y(Γ, γ) may be computed as the homology of(
C∗(E)⊗ C∗(Y(Γ, γ))
)G
. (2.3.3.4)
Recall from Theorem 2.3.2 and Remark 2.3.4 that C∗(Y(Γ, γ)) is quasi-isomorphic to a
complex which splits into a direct sum indexed by elements U ∈ Γ. The action of G is
inherited from the action of G on Γ, so as a G-module it splits as
R⊕
⊕
[U ]∈Γ/G
( ⊕
U ′∈[U ]
⊗
i∈U ′
Ĉ∗(Yl(i))
)
. (2.3.3.5)
This gives the desired splitting. We now need to show that
H∗
(
G,
⊕
U ′∈[U ]
⊗
i∈U ′
Ĉ∗(Yl(i))
) ∼= H∗(Stab(U), Ĉ∗(U)). (2.3.3.6)
To see this note that ⊕
U ′∈[U ]
⊗
i∈U ′
Ĉ∗(Yl(i)) (2.3.3.7)
is isomorphic to the Stab(U)-module Ĉ∗(U) induced up to G. Now Shapiro’s lemma gives
the desired result.
2.3.4 Homotopy quotients by automorphism groups
Let (Γ, γ, l) be a Z-labelled diagonal complex and let Y be a Z-tuple of pointed spaces.
Let Hi be a basepoint fixing group acting on Yi for each i ∈ Z. Since the diagonal complex
product is functorial with respect to the Z-tuple of factor groups there is an action of
H :=
∏
i∈Z
Hi (2.3.4.1)
46 2. DIAGONAL COMPLEXES
on Y(Γ, γ).
Proposition 2.3.8. Let (Γ, γ, l) and Y be as above and let R be a ring. Then
H∗
(
E ×H Y(Γ, γ), R
) ∼= H∗(H,R)⊕⊕
U∈Γ
H∗
(⊗
i∈Z
C∗
(
Hi, Ĉ∗(Yi)⊗Ui
))
, (2.3.4.2)
where Ui is the (perhaps empty) subset of i-coloured elements of U and the action of Hi
is via the diagonal action. When Ui is empty we take Ĉ∗(Yi)⊗Ui to be the trivial module
R.
Proof. The homology of the homotopy quotient is given be the homology of H with
coefficients in the module C∗(Y(Γ, γ)). By Remark 2.3.4 there is a quasi-isomorphism
relating C∗(Y(Γ, γ)) to a split complex (2.3.1.11). With the action of h ∈ Hi on the
summand Ĉ∗(U) given by
h.aU ′ =
haU ′ if l(U ′) = i andaU ′ otherwise, (2.3.4.3)
the quasi-isomorphism (2.3.1.11) is a morphism of H-modules. Hence the homology splits
as
H∗
(
E ×H Y(Γ, γ), R
) ∼= H∗(H,R)⊕⊕
U∈Γ
H∗
(
H, Ĉ∗(U)
)
, (2.3.4.4)
We remark that for a G1-module M and a G2-module N , the G1 × G2-coinvariants of
M ⊗N are isomorphic to the G1-coinvariants of M tensored with the G2-coinvariants of
N . Since H is a direct product of groups and Ĉ∗(U) a tensor product of modules for those
groups we get the tensor product in (2.3.4.2).
Suppose now that G is a group of automorphisms of (Γ, γ, l) and Y is a Z-tuple of
pointed spaces. Suppose also that for each z ∈ Z the discrete group Hz acts on Yz.
Then Y(Γ, γ) has both an action of G and of H. In fact together they form a semidirect
product H oG acting on Y(Γ, γ).
Combining Propositions 2.3.7 and 2.3.8 we get the main theorem.
Theorem 2.3.9. Let (Γ, γ, l), Y, H and G be as above. Then the homology of the
homotopy quotient of Y(Γ, γ) by H oG with coefficients in R splits as
H∗(H oG,R)⊕
⊕
[U ]∈Γ/G
H∗
(
H o StabG(U), Ĉ∗(U)
)
, (2.3.4.5)
where
Ĉ∗(U) :=
⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′)) (2.3.4.6)
2.3. (CO)HOMOLOGY OF PRODUCTS 47
has the permutation action by StabG(U). The action of H on C∗
(
Y(Γ, γ)
)
restricts to
Ĉ∗(U).
Proof. The action of G on Γ permutes the U ∈ Γ and so permutes the terms in the
direct sum (2.3.1.11), thus making the quasi-isomorphism (2.3.1.11) into a morphism of
G-modules. Since the action of H on C∗
(
Y(Γ, γ)
)
restricts to each term⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′)), (2.3.4.7)
the morphism from (2.3.1.11) is in fact a quasi-isomorphism of (H o G)-modules. The
homology of the homotopy quotient of Y(Γ, γ) may be calculated by the homology of
HoG with coefficients in C∗
(
Y(Γ, γ)
)
so we can compute it by calculating the homology
of H oG with coefficients in the decomposition
R⊕
⊕
U∈Γ
( ⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′))
)
. (2.3.4.8)
The direct sum ⊕
U∈Γ
(2.3.4.9)
factors into ⊕
[U ]∈Γ/G
⊕
V ∈[U ]
(2.3.4.10)
and so the (H oG)-module (2.3.4.8) decomposes over the sum⊕
[U ]∈Γ/G
. (2.3.4.11)
Now we may concentrate on individual terms in this decomposition. A term⊕
V ∈[U ]
⊗
U ′∈γ(U)
Ĉ∗(Yl(U ′)) (2.3.4.12)
may be given by inducing the HoStabG(U)-module Ĉ∗(U) up to HoG. So by the Shapiro
lemma the homology of the corresponding term is given by the homology of HoStabG(U)
with coefficients in Ĉ∗(U). This completes the proof.
Example 2.3.4.1. A special case of this theorem is when the diagonal complex comes
from a full simplicial complex ∆n = Pf [n] and so the diagonal complex product is the
direct product. With the trivial labelling this has automorphism group Sn. So for a
space Y with H the trivial group, the homotopy quotient is Y n ×Sn ESn . When Y is
a classifying space for a group G, this homotopy quotient is a classifying space for the
wreath product of G with Sn. The homology of these spaces was studied in [32].
48 2. DIAGONAL COMPLEXES
2.4 Decompositions
We are now well equipped to calculate a number of homotopical invariants of diagonal
complex products; we have a presentation for their fundamental groups and a formula for
their homology. However we know little about the higher homotopy groups, in particular
we would like to know when diagonal complex products are aspherical, that is, all their
higher homotopy groups vanish. Via the maps
Yi //Y(Γ, γ)
i //YX
pii
ii (2.4.0.13)
we can see that if any Yi has a higher homotopy group then so does Y(Γ, γ). We call
(Γ, γ) aspherical if for every labelling l : X → Z and every Z-tuple of aspherical pointed
spaces Y = (Yi)i∈Z , the diagonal complex product Y(Γ, γ) is aspherical.
We would like a combinatorial condition which would tell us which diagonal com-
plexes are aspherical, however without one we can still make progress. The following
two decompositions are tools which allow us to prove that certain diagonal complexes are
aspherical.
2.4.1 Orthogonal decomposition
This is the simpler of the decomposition theorems. Let X = A ∪ B for proper subsets A
and B of X, let (Γ, γ, l) be a Z-labelled diagonal complex on X and let Y = (Yi)i∈Z be a
Z-tuple of pointed spaces. Suppose that each U ∈ Γ is contained in at least one of A or
B. Then we say that (Γ, γ) decomposes orthogonally.
Theorem 2.4.1. Suppose that (Γ, γ) decomposes orthogonally over a set X = A ∪ B.
Then
Y(Γ, γ) = Y(ΓA, γA) ∪Y(ΓA∩B ,γA∩B) Y(ΓB, γB), (2.4.1.1)
where ΓA,ΓB and ΓA∩B are the full diagonal subcomplexes corresponding to A,B and
A ∩B respectively.
Proof. We may characterise Y(ΓA, γA) ⊆ Y(Γ, γ) as
{(yi)i∈X ∈ Y(Γ, γ) | if yi 6= ∗ then i ∈ A}. (2.4.1.2)
Then it is clear that Y(ΓA, γA)∩Y(ΓB, γB) = Y(ΓA∩B, γA∩B) and it remains to show that
Y(ΓA, γA) and Y(ΓB, γB) cover Y(Γ, γ). By Proposition 2.1.12, the product Y(Γ, γ) is
the subspace of YX covered by maps iU for U ∈ Γ. Since U ∈ Γ is either contained in A
or B, this means that iU lands in either of Y(ΓA, γA) or Y(ΓB, γB).
2.4. DECOMPOSITIONS 49
Remark 2.4.2. The reason for decomposing diagonal complex products in this way is that
one may hope to apply the Seifert van-Kampen theorem. However the natural inclusions of
the intersection into the two subspaces is not necessarily injective on fundamental groups.
Examples where the inclusions are injective are supplied by any orthogonal decomposition
of a diagonal complex coming from a simplial complex. We will see in the next chapter
in Section 3.3.3 an example where the inclusions are not injective.
2.4.2 Conical decomposition
Let x ∈ X and Γ be a proper diagonal complex on X. Suppose that for each U ∈ Γ
containing x
x+ = {x} ∈ γ(U). (2.4.2.1)
Then we say that Γ decomposes conically at x.
Theorem 2.4.3. Suppose that Γ defined over a set X decomposes conically at x. Then
there is a diagonal complex (Γlk(x), γlk(x)) and a decomposition
YΓ = YΓX−x ∪Y(Γlk(x),γlk(x))
(
Y(Γlk(x), γlk(x))× Yl(x)
)
. (2.4.2.2)
If U ∈ ΓX−x and U ∪ x+ ∈ Γ then we call U a link simplex. The link dimension of
a link simplex U is defined to be equal to the dimension of U ∪ x+ subtract one. The
following is the set of link simplices of link dimension one.
Xlk(x) =
{
U ∈ ΓX−x | U ∪ x+ ∈ Γ and γ(U ∪ x+) =
{
U, x+
}}
(2.4.2.3)
Write Γl for the set of link simplices and define γl by
γl(U) = γ(U ∪ x+)− x+ (2.4.2.4)
The pair (Γl, γl) is not necessarily a diagonal complex on X because there may exist
U ∈ Xlk(x) but not in X+.
Applying γl to U gives a partition, applying it again to the elements of γl(U) gives a
still finer partition. We repeatedly apply γl until we have reached a partition consisting of
elements of Xlk(x) at which point applying γl does not refine the partition further. Define
Û to be the subset of Xlk(x) obtained in this manner. The set Γlk(x) ⊆ PfXlk(x) is defined
to be {
Û | U ∈ Γl
}
. (2.4.2.5)
The function γlk(x) : Γlk(x) → PfΓlk(x) is defined as
γlk(x)(Û) = γ̂l(U) =
{
Û ′ | U ′ ∈ γl(U)
}
. (2.4.2.6)
50 2. DIAGONAL COMPLEXES
Lemma 2.4.4. 1. If V ⊆ U are link simplices and U ∈ Xlk(x) then V = U .
2. If V ⊆ U are link simplices then V̂ ⊆ Û .
3. The pair (Γlk(x), γlk(x)) is a diagonal complex on Xlk(x), which we call the link at x.
Proof. 1. V is a subset of U , so V ∪ x+ is a subset of U ∪ x+ and since Γ is proper
V ∪ x+ is a descendant of U ∪ x+. Hence γ(V ∪ x+) is a partial refinement of
γ(U ∪ x+) = {U, x+}. So V = U .
2. Since Γ is proper and V ⊆ U we have that V ∪ x+ is a descendant of U ∪ x+.
Hence γl(V ) is a partial refinement of γl(U). Applying this argument again to the
W ∈ γl(V ) and W ′ ∈ γl(U) with W ⊆ W ′ we have that γ2l (V ) is a partial refinement
of γ2l (U). We repeat this until we have that γ
n
l (U) = Û . Then since γ
n
l (V ) is a
partial refinement we may apply part (1) to get that it is a subset of Û and hence
that V̂ ⊆ Û .
3. Using (2) each of the required properties may be lifted from those of Γ. For example,
the simplicial condition: if V̂ is a face of Û then V is a face of U and so γl(V ) is a
partial refinement of γl(U). So using (2), γlk(x)(V ) = γ̂l(V ) is a partial refinement
of γlk(x)(U).
Let (Γst(x), γst(x)) be the diagonal complex on Xlk(x) ∪ x+ defined to consist of both U
or U ∪x+, where U are the link simplices. Then γst(x)(U) = γlk(x)(U) and γst(x)(U ∪x+) =
γlk(x)(U) ∪ {x+}. This is an example of a direct product of diagonal complexes and
Y(Γst(x), γst(x)) ∼= Y(Γlk(x), γlk(x))× Yl(x).
Proof of Theorem 2.4.3. The subspace YΓX−x ⊆ YΓ is characterised by
{(yi)i∈X ∈ YΓ | yx = ∗}. (2.4.2.7)
Meanwhile the subspace YΓst(x) ⊆ YΓ is characterised by
{(yi)i∈X ∈ YΓ | the set {i | yi 6= ∗} − x is a link simplex}. (2.4.2.8)
And so their intersection consists of those elements whose support is a link simplex.
So by the definition of (Γlk(x), γlk(x)) the intersection is Y(Γlk(x), γlk(x)). To prove the
amalgamation we just need to note that any U ∈ Γ either does not contain x or U − x is
a link simplex, so the two spaces do cover Y(Γ, γ).
Finally we have that
Y(Γst(x), γst(x)) = Y(Γlk(x), γlk(x))× Yl(x). (2.4.2.9)
2.5. COSET COMPLEXES 51
Example 2.4.2.1. Let Γ be the infinite two dimensional diagonal complex from Exam-
ple 2.1.5.4. We saw in Example 2.1.6.2 that Γ is the colimit of a filtration by Γn, which is
given in Equation (2.1.6.11). The diagonal complex Γn is the disjoint union of an infinite
number of points {n+ 2, . . .} and a finite diagonal complex on {1, . . . , n+ 1} which we
denote by Γ′n. The filtration
Γ′1 < Γ
′
2 < . . . (2.4.2.10)
has colimit Γ and is injective on fundamental group, so to show that Γ is aspherical it
is enough to show that Γ′n is aspherical for each n. We do this by induction using the
conical decomposition, the diagonal complex Γ′1 is just a line so is aspherical, now assume
that Γ′n−1 is aspherical for some n ≥ 2. The diagonal complex Γ′n decomposes conically
at n+ 1; the star at n+ 1 is a line and the link is a point. The inclusion of the link space
is injective on fundamental group so the conical decomposition allows us to apply 1.1.1
to see that Γ′n is aspherical. Now by induction Γ
′
n is aspherical for every n and hence so
is Γ.
2.5 Coset complexes
In the previous section we introduced two decompositions as a means to study diagonal
complexes as iterated pushouts. In this section we introduce a different technique which
applies to diagonal complex products of groups. The approach is to consider certain
coset complexes associated to natural subgroups. The notion of coset complex we use
was defined in [1] although this name isn’t used. The coset complex of a finite group is
studied in [8] as the simplicial complex associated to the coset poset.
Recall that the diagonal complex product of groups is defined as the colimit from a
certain category PΓ, see Section 2.1.7. Each object {Ui} of PΓ determines a subgroup∏
Uj∈{Ui}
Gl(Uj) ↪→ G(Γ, γ), (2.5.0.11)
the fact that this morphism is injective can be seen by taking the composition∏
Uj∈{Ui}
Gl(Uj) → G(Γ, γ)→
∏
x∈X
Gl(x), (2.5.0.12)
which is a diagonal map and hence injective. The category PΓ was defined as a subposet
closed under meets of the partial partitions of X under partial coarsening. The meet in
this poset corresponds to the intersection of subgroups in G(Γ, γ), so the collection of
subgroups parametrised by PΓ is closed under intersections.
We now recall some material from the paper [1] on which our treatment of the coset
complex is based. Let G be a group with a finite family of subgroups H = {Hj | j ∈ J}
52 2. DIAGONAL COMPLEXES
closed under intersection. For such a group let H be the set of cosets∐
j∈J
G/Hj. (2.5.0.13)
Since H is closed under taking intersections and since a non-empty intersection of cosets
g1H1∩g2H2 is a coset of the intersection of the respective subgroups u(H1∩H2), the cosets
H are closed under taking non-empty intersections. The set H may be viewed as a cover
of G.
Let X be a set and U be a covering of that set, we will assume that U is closed under
taking non-empty intersections. Under inclusion U forms a poset (U,⊆), which has a
nerve N(U,⊆) which is a simplicial set where the k-simplices are chains
U0 ⊆ U1 ⊆ . . . ⊆ Uk. (2.5.0.14)
We also define another simplicial set Xsimp(U) where the k-simplices are (k + 1)-tuples
(x0, . . . , xk) of X such that there is a U in U containing each xi for i = 1, . . . , k. Re-
sults from Section 1.6 and Theorem 1.4 from [1] imply that Xsimp(U) and N(U,⊆) are
homotopic.
Definition 2.5.1. Let G be a group with a finite family of subgroups H = {Hj | j ∈ J}
closed under intersection and let H be the associated set of cosets which may be viewed
as a cover of G. Then the nerve N(H,⊆) is called the coset complex.
The functor B assigns to a group G the standard simplicial set B(G), a classifying
space for G. Recall that the set of k-simplices of B(G) is the set Gk.
Theorem 2.5.2. Let G be a group with a finite family of subgroups H = {Hj | j ∈ J}
closed under intersection. Let (H,⊆) be the poset of cosets as above. Then the colimit
colimH∈HB(H) is a classifying space for G if and only if the coset complex N(H,⊆) is
contractible.
Proof. For a group H, define Xsimp(H) to be the simplicial set with k-simplices the (k+1)-
tuples of H. The face maps are given by forgetting a coordinate and the degeneracy maps
given by duplicating coordinates. This has a free H-action given by
h.(h0, . . . , hk) = (hh0, . . . , hhk). (2.5.0.15)
This construction is functorial so if H is a subgroup of G then there is an inclusion of
simplicial sets Xsimp(H) ↪→ Xsimp(G). Define Xsimp(H) ↑G to be the G-orbit of the image
of Xsimp(H) in Xsimp(G). The induction notation is appropriate because both H and G
actions are free. Explicitly the k-simplices of Xsimp(H) ↑G consist of those (k + 1)-tuples
2.6. QUESTIONS AND DISCUSSION 53
which lie in a single coset gH. Note that if H ≤ H ′ then Xsimp(H) ↑G⊆ Xsimp(H ′) ↑G
and furthermore
Xsimp(H1) ↑G ∩Xsimp(H2) ↑G= Xsimp(H1 ∩H2) ↑G . (2.5.0.16)
So the colimit colimH∈HXsimp(H) ↑G is a subspace of Xsimp(G). This subspace is the span
of all of the inclusions of Xsimp(H) ↑G for H ∈ H, so explicitly it consists of (k+1)-tuples
(g0, . . . , gk) for which there exists a coset gH containing each gi for i = 0, . . . , k. But this
is precisely the space Xsimp(H). The space Xsimp(H) has a free G-action and we now take
the quotient:
Xsimp(H)/G = colimH∈HXsimp(H) ↑G /G ∼= colimH∈HXsimp(H)/H. (2.5.0.17)
Each Xsimp(H) is a contractible space with a free H-action and Xsimp(H)/H ∼= B(H),
this means that
Xsimp(H)/G ∼= colimH∈HB(H). (2.5.0.18)
So colimHB(H) is a classifying space for G if and only if Xsimp(H) is contractible. However
Xsimp(H) is homotopic to N(H,⊆). We are done.
This may be applied to a diagonal complex product as follows. Let (Γ, γ) be a Z-
labelled diagonal complex and let G be a Z-tuple of groups. As discussed above the
category PΓ parametrises a family of subgroups of G(Γ, γ) which are closed under in-
tersections. This satisfies the conditions of Theorem 2.5.2 and we write CCΓ(G) for the
coset complex of this family.
2.6 Questions and discussion
We now take stock of our progress in the study of diagonal complexes. Each Z-labelled
diagonal complex supplies a product, which can be seen as a functor from CZ to C where C
is a category with finite products and colimits. The diagonal complex product is defined
as the colimit of a diagram and in the case of pointed spaces this diagram comes from a
covering of the product by subspaces.
Every diagonal complex comes with a levelwise filtration and to construct a diagonal
complex product one may glue on pieces level by level. This is used to compute the diag-
onal complex product of groups and the (co)homology of the diagonal complex product
of pointed spaces. Recall that the fundamental group of a product of pointed spaces is
the product of the fundamental groups.
After Sections 2.2 and 2.3 we have a good description of the fundamental group and
(co)homology of a diagonal complex product of pointed spaces. The major gap in our
understanding is the higher homotopy groups of diagonal complex products. In particular
54 2. DIAGONAL COMPLEXES
we would like to know when a diagonal complex is aspherical, recall that this means that
the associated product of aspherical spaces is itself aspherical. The problem with higher
homotopy groups is that general colimits are very difficult to calculate and using the level-
wise construction is not helpful. However in Section 2.4 we describe two decompositions,
the main point of these is to write a complicated product as a pushout of three simpler
products. Unfortunately we can not assume that the inclusions from the intersection are
injective on fundamental groups but often we are in a position to check the injectivity.
This is important because although colimits are in general hard to compute, the pushout
of aspherical spaces with injective maps is itself aspherical.
In the next chapter we will use the decompositions to show that particular examples
of diagonal complexes are aspherical, however we will also see an example where the
injectivity fails. In this example we use the coset complex, which is an alternative method
for proving that diagonal complexes are aspherical.
Our aim should be to have a combinatorial condition which would tell us which diag-
onal complexes are aspherical. The following case would be an important step towards
the general case.
Question A. Let Γ be a proper diagonal complex on a set X. Suppose that X ∈ Γ, then
is Γ aspherical?
When working with right-angled Artin groups, which are the graph products of the
infinite cyclic group Z, one uses CAT(0) geometry to show that the graph product of the
circle S1 is aspherical. The following diagonal complex is an example where the product
of copies of S1 is not CAT(0), but is aspherical.{
{1, 2, 3}, {2, 3, 4},
{1, 2}, {3, 4}
}
∪X+ (2.6.0.19)
Even when the spaces are not CAT(0) we may hope that geometrical techniques such as
Morse theory could be helpful. However these apply only directly to products of S1 and
so an answer to the next question would mean they are helpful for general products.
Question B. Let Γ be a proper diagonal complex on a set X and let Y1 be the unit circle.
If Y(Γ, γ) is aspherical, does this imply that Γ is aspherical?
Chapter 3
Symmetric automorphisms of free
products
We now get onto the main topic of this thesis; automorphisms of free products. In
Section 3.1 the different types of automorphism we are interested in are introduced. At
the center of our methodology is a classifying space for a group of partial conjugations
(introduced shortly). This space is most naturally described geometrically as a moduli
space of certain diagrams which we name cactus products, see Section 3.2. The paths in
this moduli space give a geometric interpretation for partial conjugations.
The previous chapter was devoted to diagonal complexes, which we will put to use
in Section 3.3. We show that the moduli space of cactus products may be written as a
diagonal complex product, associated to a particular diagonal complex of forest posets.
Using this description we may prove the main theorems of this chapter and so compute
the (co)homology of the symmetric automorphism groups in Section 3.4.
There is another interpretation of the partial conjugations; this time as the automor-
phisms which are extendable with respect to the free product functor. We will say what
this means in Section 3.5. Then in Section 3.6 the relationship with Outer space will be
discussed. Finally in Section 3.7 we give generalisations of the diagonal complex of forest
posets, then conjecture that the associated groups describe the symmetric automorphisms
of graph products of groups.
3.1 Introduction
Let G be an n-tuple of groups and let G be their free product. We are interested in
automorphisms of this G. Example automorphisms include:
Example 3.1.0.2. 1. Let φ ∈ Aut(G1) be an automorphism of G1. Then
G1 ∗G2 ∗ . . . ∗Gn
φ∗1G2∗...∗1Gn // G1 ∗G2 ∗ . . . ∗Gn (3.1.0.1)
55
56 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
is an automorphism of G which we call a factor automorphism. The original auto-
morphism could have been of any Gi.
2. Suppose that ϕ : G1 → G2 is an isomorphism. Let τ12 be the map which switches
G1 and G2 like so
G1 ∗G2 ∗ . . . ∗Gn τ12 // G2 ∗G1 ∗ . . . ∗Gn, (3.1.0.2)
then composing this with
G2 ∗G1 ∗ . . . ∗Gn ϕ
−1∗ϕ∗...∗1Gn// G1 ∗G2 ∗ . . . ∗Gn, (3.1.0.3)
gives an automorphism of G. There is ofcourse an analogue for each pair {i, j}.
3. Let i 6= j and h be an element of Gj. Define an automorphism of G by
αhi : gk 7→
h−1gkh if k = i and gk ∈ Gkgk if k 6= i and gk ∈ Gk. (3.1.0.4)
This is called a partial conjugation of G.
Remark 3.1.1. If we let i = j then Equation 3.1.0.4 defines a factor automorphism induced
by the inner automorphism inn(h) of Gi.
Let FR(G) be the subgroup of Aut(G) generated by the partial conjugations. Then
it is simple to check that the following relations hold:
αhi .α
g
i = α
gh
i for g, h ∈ Gj, (3.1.0.5)
α
ej
i = e for ej the identity in Gj, (3.1.0.6)
αhi .α
h′
k = α
h′
k .α
h
i , (3.1.0.7)
where h ∈ Gj and h′ ∈ Gj′ with i 6= k and j, j′ /∈ {i, k}. The last relation says that if
automorphisms ‘act on different factors’ then they commute, note that j could be equal
to j′. The next relation
αhi .α
h
k .α
t
i = α
t
i.α
h
i .α
h
k , (3.1.0.8)
where h ∈ Gj and i, j and k are distinct and t ∈ Gk, requires checking: first the left hand
side
αhi .α
h
k .α
t
i(gi) = α
h
i .α
h
k(g
t
i) = α
h
i (g
h−1th
i ) = g
th
i (3.1.0.9)
(gk) = α
h
i α
h
k(gk) = g
h
k (3.1.0.10)
(gl) = gl for l 6= i, k. (3.1.0.11)
3.1. INTRODUCTION 57
Now for the right hand side:
αti.α
h
i .α
h
k(gi) = α
t
i.α
h
i (gi) = α
t
i(g
h
i ) = g
th
i (3.1.0.12)
(gk) = α
t
iα
t
i(g
h
k ) = g
h
k (3.1.0.13)
(gl) = gl for l 6= i, k. (3.1.0.14)
So the relation holds. We will give a geometric interpretation of these automorphisms in
Section 3.2. The following proposition is taken from the work of Collins and Zieshang [15],
[16]. It generalises the presentation of the symmetric automorphism group given by
McCool [36].
Proposition 3.1.2. The relations (3.1.0.5), (3.1.0.6), (3.1.0.7) and (3.1.0.8) along with
the generators αhi give a presentation for FR(G).
Recall from Example 3.1.0.2 that each automorphism group Aut(Gi) acts on the free
product G. If i 6= j then the automorphism groups act on different factors and so the
actions commute. Hence we have a subgroup
Aut(G) =
n∏
i=1
Aut(Gi) (3.1.0.15)
of the automorphism group Aut(G). Now let h ∈ Gj and αhi be a generating partial
conjugation. Let φ ∈ Aut(Gk). If k 6= j then φ commutes with αhi . Now suppose that
k = j then
φαhi φ
−1 = αφ(h)i (3.1.0.16)
as can be checked by looking at the action of the left hand side on the factors of G. Hence
as a subgroup of Aut(G), the partial conjugations and the factor automorphisms generate
a semi-direct product
FR(G)oAut(G). (3.1.0.17)
We denote this by PAut(G) and call it the pure symmetric automorphism group of G.
Now let Inn(G) be the subgroup
n∏
i=1
Inn(Gi) (3.1.0.18)
of Aut(G) consisting of the automorphisms of G induced by inner automorphisms of
the factors. Note that this subgroup is not necessarily inner itself, and does not usually
contain the inner automorphisms of G either. The subgroup of Aut(G) generated by
Inn(G) and FR(G) is called the Whitehead group of G and is denoted by
WH(G) = FR(G)o Inn(G). (3.1.0.19)
58 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
Consider the n-tuple of groups G = (Gi) and divide it into equivalence classes under
isomorphism. For each equivalence class (Gi)i∈A choose a representative Gi and for each
j ∈ A − {i} choose an isomorphism ϕj : Gi → Gj, also let ϕi be the identity. Now each
pair of isomorphic groups Gj, Gk may be identified uniquely by
Gj
ϕ−1j // Gi
ϕk // Gk. (3.1.0.20)
Using this isomorphism in 2) of Example 3.1.0.2 we get an automorphism of G. All such
automorphisms generate a subgroup
SG ∼= Sn1 × . . .×Snk , (3.1.0.21)
where ni are the sizes of the equivalence classes. Note that different choices for the
isomorphisms φj would have given a different embedding of (3.1.0.21) into Aut(G).
Let σ ∈ SG and h ∈ Gj with αhi an associated partial conjugation. Then
σαhi σ
−1 = α
ϕσ(j)ϕ
−1
j (h)
σ(i) . (3.1.0.22)
For convenience we will write h for ϕσ(j)ϕ
−1
j (h) using ϕσ(j)ϕ
−1
j to identify the sets Gi and
Gσ(i). The subgroup generated by SG and FR(G) is a semidirect product
ΣFR(G) = FR(G)oSG (3.1.0.23)
and is called the symmetric Fouxe-Rabinovitch group. The group SG also acts by conju-
gation on Aut(G) and Inn(G), so we may define subgroups
ΣWH(G) = WH(G)oSG and (3.1.0.24)
ΣAut(G) = PAut(G)oSG, (3.1.0.25)
which are called the symmetric Whitehead group of G and the symmetric automorphism
group of G respectively. The following proposition is derived from the Kurosh subgroup
theorem.
Proposition 3.1.3. Let G be an n-tuple of groups such that each Gi is freely indecom-
posable and not isomorphic to the infinite cyclic group. Then
ΣAut(G) ∼= Aut(G). (3.1.0.26)
3.2. MODULI SPACES OF CACTUS PRODUCTS 59
Summarising we have a diagram of subgroups of Aut(G):
ΣFR(G) // ΣWH(G) // ΣAut(G)
FR(G) //
OO
WH(G) //
OO
PAut(G).
OO
(3.1.0.27)
Every arrow is a normal embedding as are their composites. Factoring out by the normal
subgroup FR(G) gives the corresponding diagram.
SG // Inn(G)oSG //Aut(G)oSG
I //
OO
Inn(G) //
OO
Aut(G).
OO
(3.1.0.28)
Most importantly we have that
ΣAut(G) ∼= FR(G)oAut(G)oSG (3.1.0.29)
We have omitted the bracketing deliberately, either semi-direct product, o may be eval-
uated first.
3.2 Moduli spaces of cactus products
In the category of pointed spaces the analogue of the n-fold free product of groups is the
n-fold wedge sum; this takes n pointed spaces and identifies their basepoints.
Y2 Y3Y1 (3.2.0.30)
The fundamental group of a wedge sum is the free product of the fundamental groups of
the summands. The assumption that the spaces have the homotopy type of CW complexes
means that changing the basepoint does not alter the homotopy type1, so forming the
wedge sum with a different choice of basepoints gives the same homotopy type. The
moduli space of cactus products describes a space of possible ways to ‘wedge’ n spaces
together.
3.2.1 Cactus products
Let t be a tree with vertex set [n] = {1, . . . , n}. A tree t with a chosen vertex i is called a
rooted tree. The edges of a rooted tree may be oriented by pointing them towards the root,
1we are assuming that all our spaces are connected
60 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
for an edge e the source vertex is denoted s(e) and the target vertex t(e). An example of
a rooted tree with root 4:
2
=
==
= 1
5
=
==
= 3
4 .
(3.2.1.1)
Now let (Yi, ∗i) be pointed spaces for i = 1, . . . , n. A cactus diagram T over rooted tree
t consists of an edge labelling of t: to each edge e there is a label ye ∈ Yt(e). A cactus
diagram T gives a cactus product space YT :
Y1 q . . .q Yn
∗s(e) ∼ ye | e ∈ t . (3.2.1.2)
An example corresponding to the rooted tree (3.2.1.1) above:
Y4
Y5
Y2
Y3
Y1
(3.2.1.3)
If each pointed space is homotopy equivalent to a CW complex then the cactus product
space is homotopic to the wedge product Y1 ∨ . . . ∨ Yn. A cactus product space has a
canonical choice of basepoint given by the basepoint of the space corresponding to the root
of the tree. For each i there is a natural inclusion Yi ↪→ YT , but note that this inclusion
does not respect the basepoint. The cactus product consists of the cactus product space
in the context of the diagram
YT
Y1
;;vvvvv · · · · · · Yn.
ddIIIII (3.2.1.4)
3.2.2 The moduli space of cactus products
Let T and T ′ be two cactus products with the same number of spaces. Then we say that
T and T ′ are congruent if there is a map YT → YT ′ making the diagram
YT // YT ′
Yi
==||||
``AAAA (3.2.2.1)
3.2. MODULI SPACES OF CACTUS PRODUCTS 61
commute for each i. For example the two decorated trees
2
y =
==
= 3
y
1
and
3
∗
2
y
1
(3.2.2.2)
give the same space:
Y2 Y3
Y1
y (3.2.2.3)
Lemma 3.2.1. For a fixed tree t, a cactus product YT determines the labelling T . And
so the set of cactus products over t is naturally identified with∏
e
Yt(e). (3.2.2.4)
Proof. Given an edge e the intersection
Ys(e) ∩ Yt(e) ↪→ Yt(e). (3.2.2.5)
consists of a single point, which is the label ye ∈ Yt(e).
The moduli space of cactus products MY is defined to be the set of cactus prod-
ucts modulo congruence, with the topology inherited from the CW product topology
of (3.2.2.4). It has a canonical basepoint given by the wedge product, which is realised
by any tree with every label a basepoint.
3.2.3 An embedding into a direct product
Proposition 3.2.2. The moduli space of cactus graphs MY may be embedded into
Y n−11 × . . .× Y n−1n . (3.2.3.1)
Proof. A coordinate in the product space is denoted
(yij)i 6=j, (3.2.3.2)
where i, j = 1, . . . , n and yij is an element of Yi. Let T be a cactus diagram over a tree t.
For each pair (i, j) choose a path within the cactus product YT from the basepoint of Yj
to the basepoint of Yi. Let yij be the element in Yi first reached by the path.
62 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
Combinatorially this may be defined by taking the unique undirected path in t from
j to i. If each edge of the path points towards the root, then define yij to be ye where e
is the last edge in the path (and so has end vertex i). If the path goes through the root
then let yij be the basepoint ∗ ∈ Yi.
This map is both well-defined and injective.
3.2.4 Geometric interpretation of partial conjugations
A based loop in the moduli space of cactus products of pointed spaces Y1, . . . , Yn can be
seen to act on the free product of groups
pi1Y1 ∗ . . . ∗ pi1Yn. (3.2.4.1)
This is illustrated by the following diagram which shows a path representing the auto-
morphism αg2.
Y2
Y1 g* (3.2.4.2)
Any loop γ contained in Y2 is taken to a loop which first follows g in Y1 and then follows
γ before returning around g in the opposite direction. The commutator relations between
partial conjugations can be viewed as embedded tori inside the moduli space, for example
the following represents such a torus:
Y2
Y4g*
Y1
Y3
h (3.2.4.3)
The factor space Y1 may pass around loop h independently to space Y2 passing around
loop g. The torus pictured on the right records a homotopy between paths representing
3.3. THE DIAGONAL COMPLEX OF FOREST POSETS 63
αh1α
g
2 and α
g
2α
h
1 . This next diagram represents a different type of torus:
Y1g*
Y3
Y2 h
(3.2.4.4)
One edge of the torus has space Y3 passing around loop h, which represents the auto-
morphism αh3 . The other edge has both spaces Y2 and Y3 passing around loop g, this
represents the automorphism αg2α
g
3. Therefore this torus represents a homotopy between
paths representing αg2α
g
3 and α
h
3 .
3.3 The diagonal complex of forest posets
The moduli space of cactus products has an intuitive description, but when it comes
to working hands on with the space; decomposing it into smaller pieces, calculating its
fundamental group and calculating its homology, an alternative description using diagonal
complexes is far more powerful. The diagonal complex that does this consists of a set of
different partial orders on a fixed set [n]. The partial orders of interest are those which
give a Hasse diagram of planted forests, hence the name forest posets.
3.3.1 The construction of ΓFn
Let (P,≤) be a finite poset. The Hasse diagram of (P,≤) is the directed graph with
vertex set P and an edge
−→
ij when i < j are adjacent, that is, when i ≤ k ≤ j implies that
k = i or j. Conversely a directed graph without directed loops defines a poset given by
setting i < j when
−→
ij is an edge and then taking the transitive closure. A directed graph
is called a planted forest if for each vertex the number of incoming edges is not greater
than one, if there are no cycles and if the edge set is non-empty. Under the transitive
closure the planted forests are taken to posets (P,≤) with the underset condition:
for all x ∈ P, {y | y ≤ x} is a total order. (3.3.1.1)
In fact the correspondence is a bijection between the set of planted forests and the set of
finite non-trivial posets satisfying (3.3.1.1). Let Pn be the set [n]. A poset (Pn,≤) defines
64 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
a set
{(i, j) | i < j}, (3.3.1.2)
which is a subset of Xn = Pn×Pn−∆Pn. We write ΓFn ⊆ PfXn for the set of subsets of
Xn given by forest posets (Pn,≤). An uppercase letter such as U will be used to denote
the poset (Pn,≤), the subset of Xn and the corresponding planted forest. This should
not cause confusion.
We will next define a map γFn : ΓFn → PfΓFn . For a forest poset U take i < j, then
choose a maximal path from i to j
i = i0 < i1 < . . . < im = j (3.3.1.3)
The pair (i, i1) necessarily gives an edge
−→
ii1 ∈ U , write µU(i, j) = −→ii1. Then µU is a map
from the set U to E(U), the edge set of the planted forest. Since the path is within the
set {k | k ≤ j} which is a total order, the path is unique and so µU is well-defined. For
an edge
−→
ij we have µU(i, j) =
−→
ij , so µU is surjective. The map µU defines a partition of
U , each subset is given in the form µ−1U (
−→
ij ) for some edge of U , and this defines γ(U).
For example
γFn
1
2
OO
4
3
OO @@
=
{
1
2
OO
,
1 2
3
OO @@ ,
3
4
OO
}
. (3.3.1.4)
Proposition 3.3.1. The pair (ΓFn , γFn) is a diagonal complex on Xn. Furthermore this
diagonal complex is proper.
Proof. For each element (i, j) of Xn = Pn×Pn−∆Pn, the tree i //j gives the poset given
by {(i, j)} and so condition (1) in the definition of a diagonal complex is satisfied. That
γ(U) is a partition of U is immediate from the definition. If γ(U) = {U} then U is a
forest with only one edge and so is contained in X+n , hence condition (2) is satisfied.
Next we must show that ΓFn contains the faces of each U ∈ ΓFn . Let {Ui}i∈E(U) = γ(U)
and suppose Z ⊆ E(U). We write UZ for
⋃
i∈Z Ui. Then UZ is the subset of U consisting
of i >
0
OO ==
0
OO <<
0
OO
2 21 215 2150 21502
(3.4.1.3)
This algorithm may be reversed in order to build a tree from a word and this produces
the bijection. The element
(x0 + x1 + . . .+ xn)
n−1 (3.4.1.4)
in the free commutative ring on {x0, . . . , xn} is the sum over all words of length n − 1.
The word associated with a planted forest may be turned into a monomial by viewing it
3.4. (CO)HOMOLOGY OF THE SYMMETRIC AUTOMORPHISM GROUPS 77
in the free commutative monoid on {x1, . . . , xn} and setting x0 = 1. Since the number
of xi’s in a word is the number of outgoing edges of i, this monomial is the same as the
monomial defined in (3.4.1.1). Hence setting x0 = 1 and viewing (3.4.1.4) in the free
commutative ring gives the Hilbert-Poincare´ series of the planted forests.
The series for the planted forests is identical to the Hilbert-Poincare´ series of the
diagonal complex describing (G1 ∗ . . . ∗Gn)×n−1 and so by Corollary 2.3.5:
Theorem 3.4.2. Let G be an n-tuple of groups and G be the n-fold free product of these
groups. Then
H∗(FR(G), R) ∼= H∗(Gn−1, R), (3.4.1.5)
where Gn−1 is the (n− 1)-fold direct product of G. This also holds for the cohomology.
The Euler characteristic of WH(Z∗n) ∼= FR(Z∗n), was computed in [35], and the
homology was calculated in [27]. Recalling that the Hilbert-Poincare´ series of Z is 1 + t
we may reobtain these results.
Corollary 3.4.3. The Hilbert-Poincare´ series of H∗
(
WH(Fn),Z
)
is
(1 + tn)n−1 (3.4.1.6)
and so the Euler characteristic is
(1− n)n−1. (3.4.1.7)
We may take this further by calculating the homology of WH
(
Z/(p)∗n
)
. The question
of the cohomology of this group was posed by Jensen in [39].
Corollary 3.4.4. The Hilbert-Poincare´ series of H∗
(
WH(Z/(p)∗n),Z
)
is
1 + y
1
1 + t
[(
1 +
nt
1− t
)n−1
− 1
]
, (3.4.1.8)
where the y coefficient encodes the number of Z/(p)-summands and the constant term
encodes the number of Z-summands.
Proof. The homology of Z/(p) is given by
Hi(Z/(p),Z) =
Z if i = 0,
Z/(p) if i = 2k − 1 for k ≥ 1 and
0 if i = 2k for k ≥ 1.
(3.4.1.9)
As a Hilbert-Poincare´ series this is
1 + yt+ yt3 + yt5 + . . . = 1 +
ty
1− t2 . (3.4.1.10)
78 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
Putting this into the Hilbert-Poincare´ series for ΓFn gives(
1 +
nyt
1− t2
)n−1
= 1 +
n−1∑
k=1
(
n− 1
k
)(
nyt
1− t2
)k
. (3.4.1.11)
Using the identity y2 = y(1 + t) from (2.3.1.16) to give yk = y(1 + t)k−1 and simplifying
we get the desired series (3.4.1.8).
3.4.2 The pure automorphism groups
Recall that the pure automorphism group, PAut(G) consists of those automorphisms
that take an element g ∈ Gi and give a conjugate (g′)h of an element g′ ∈ Gi inside
G = G1 ∗ . . . ∗Gn . It is a semidirect product of FR(G) and Aut(G) ∼=
∏
i Aut(Gi). The
action of Aut(G) on FR(G) may be described as follows. Let φ ∈ Aut(Gi), then
(α
gj
k )
φ =
α
gj
k if j 6= i and
α
φ(gj)
k if j = i.
(3.4.2.1)
The group Aut(G) has an action on the free product G and hence on Gn−1 by the diagonal
action, so we may form the group
Gn−1 oAut(G). (3.4.2.2)
Theorem 3.4.5. The homology of PAut(G) decomposes as
H∗
(
Aut(G), R
)⊕ ⊕
F∈ΓFn
H∗
(
Aut(G), Ĉ∗(F )
)
, (3.4.2.3)
where
Ĉ∗(F ) =
⊗
−→
ij∈E(F )
Ĉ∗(Yi) (3.4.2.4)
and Yi is a classifying space for Gi for each i = 1, . . . , n.
Therefore
H∗
(
PAut(G), R
) ∼= H∗(Gn−1 oAut(G), R). (3.4.2.5)
Proof. The diagonal complex product YΓFn is a classifying space for FR(G) and has
an action of Aut(G). The homotopy quotient is then a classifying space for the pure
automorphism group, PAut(G) = FR(G)oAut(G).
Let ΓFn have the labelling by [n] given by l(i, j) = i. For each i ∈ [n] let Hi = Aut(Gi)
be the automorphism group of G. The direct product, H of the Hi acts on YΓFn see
Section 2.3.4. This direct product is isomorphic to Aut(G) and has the same action on
3.4. (CO)HOMOLOGY OF THE SYMMETRIC AUTOMORPHISM GROUPS 79
YΓFn . Letting S be the trivial group, then we may apply Theorem 2.3.9 to calculate the
homology of PAut(G), which gives the decomposition.
For the second part observe that for any planted forest F ∈ ΓFn with the same mono-
mial as a term U in the homology of Gn−1, the automorphism groups always act diagonally
and so Ĉ∗(F ) ∼= Ĉ∗(U) as H-modules. Since the Hilbert-Poincare´ polynomials are identi-
cal the corresponding sums of H-modules are isomorphic. Hence applying Theorem 2.3.9
for both groups PAut(G) and Gn−1 oAut(G) we get the same homology.
3.4.3 The symmetric automorphism groups
Let n1, . . . , nk be the sizes of the isomorphism classes of G = (G1, . . . , Gn). Then write
SG for the group
Sn1 × . . .×Snk ≤ Sn (3.4.3.1)
of symmetries of G. The symmetric automorphism group ΣAut(G) is the semidirect
product of the pure automorphism group PAut(G) and SG.
Remember that the diagonal complex ΓFn has a universal labelling Xn → [n]. So a
map L : [n]→ Z = [k] induces a labelling l : Xn → Z. Let ni be the size of the set L−1(i)
for i ∈ Z. We will call a planted forest Z-coloured if there is a vertex colouring by Z and
if for each i ∈ Z there are ni vertices coloured i.
Lemma 3.4.6. The {1}-labelled diagonal complex ΓFn carries an action of Sn. The orbits
are given by the isomorphism types of unlabelled planted forests.
The Z-labelled diagonal complex ΓFn carries an action of SG = Sn1 × . . .×Snk . The
orbits are given by the isomorphism types of Z-coloured planted forests.
Proof. The symmetric group Sn has the permutation action on [n]. This gives an action
on Xn = {(i, j) | i 6= j}. The induced action on PfXn restricts to ΓFn : an element σ ∈ Sn
takes the edge
−→
ij of U ∈ ΓFn to
−−−−−→
σ(i)σ(j) of σ(U).
The action of SG on [n] fixes the morphism L : [n]→ Z and hence the action on ΓFn
satisfies
l(σ(U)) = l(U), (3.4.3.2)
for all σ ∈ SG and U ∈ ΓFn .
Example 3.4.3.1. Let L : {1, 2, 3} → {◦, •} be defined by l(1) = l(2) = ◦ and l(3) = •.
Then the orbits are enumerated by the Z-coloured planted forests as follows:
•
◦
OO
◦
OO
◦
•
OO
◦
OO
◦
◦
OO
•
OO
◦ ◦
•
OO >>~~~
• ◦
◦
OO >>~~~
◦
•
OO
◦
•
◦
OO
◦
◦
◦
OO
.• (3.4.3.3)
80 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
Theorem 3.4.7. Let G = G1∗. . .∗Gn where Gi is neither freely decomposable nor infinite
cyclic. Then the homology of the automorphism group of G is given by
H∗
(
Aut(G), R
) ∼= H∗(Aut(G)oSG, R)⊕ ⊕
f∈ForestsZ
H∗
(
Aut(G)oAut(f), Ĉ∗(f)
)
,
(3.4.3.4)
where ForestsZ is the set of Z-coloured planted forests and Ĉ∗(f) is given by⊗
i∈[n]
Ĉ∗(Yl(i), R)⊗out(i), (3.4.3.5)
where Yi is a classifying space for Gi for each i = 1, . . . , n.
Proof. There are two labellings of ΓFn , the labelling by Z coming from the isomorphism
type of the Gi and the universal labelling by [n]. The group SG acts on ΓFn preserving
the first labelling. Thus we may apply Theorem 2.3.9 to compute the homology of the
homotopy quotient of the action of Aut(G) oSG on YΓFn . This homotopy quotient is
a classifying space for ΣAut(G).
By Lemma 3.4.6 the orbits of ΓFn are given by ForestsZ and the stabiliser of a
Z-coloured forest f is Aut(f).
Remark 3.4.8 (cohomological version of Theorem 3.4.7). Precisely the same arguments
may be used to prove that given the hypotheses of Theorem 3.4.7 the cohomology of
Aut(G) is given by
H∗
(
Aut(G), R
) ∼= H∗(Aut(G)oSG, R)⊕ ⊕
f∈ForestsZ
H∗
(
Aut(G)oAut(f), Ĉ∗(f)
)
,
(3.4.3.6)
where Ĉ∗(f) is given by ⊗
i∈[n]
Ĉ∗(Yl(i))⊗out(i). (3.4.3.7)
Remark 3.4.9. Let G = G∗n11 ∗ . . . ∗G∗nkk with each Gi neither freely decomposable nor Z,
then Aut(G) ∼= PAut(G)oSG, so we have that the Euler characteristic of Aut(G) is
1
(n1)! . . . (nk)!
(
1 + n1(χ(G1)− 1) + . . .+ nk(χ(Gk)− 1)
)n−1 k∏
i=1
χ
(
Aut(Gi)
)−ni , (3.4.3.8)
which is the same as that of (Gn−1 oAut(G))oSG. However their integral homologies
3.5. CATEGORICAL INTERPRETATION OF SYMMETRIC AUTOMORPHISMS 81
are not necessarily the same due to the presence of the groups Aut(f). Hence the pattern
H∗(FR(G), R) ∼= H∗(Gn−1, R), (3.4.3.9)
H∗(WH(G), R) ∼= H∗(Gn−1 o Inn(G), R), (3.4.3.10)
H∗(PAut(G), R) ∼= H∗(Gn−1 oAut(G), R) (3.4.3.11)
is broken when non-pure symmetric automorphisms are present.
Remark 3.4.10. There is a discrepancy between the three isomorphisms we obtain in
(3.4.3.9), (3.4.3.10) and (3.4.3.11) and the results of [28] and [4]. Let G = G1 ∗ . . . ∗ Gn
be a free product with n factors. Calculating the Euler characteristic we find that
χ(FR(G)) = χ(Gn−1), (3.4.3.12)
whereas in [28] it is calculated to be
χ(Gn−1).
∏
i
χ(Inn(Gi))
−1
. (3.4.3.13)
Across all of the results there is a factor of χ(Inn(G)) difference. There is an analogous
difference with the results of [4]. We attribute this to a misquoting of Proposition 5.1
of [37] in each of the papers.
Example 3.4.3.2. Let Gi = Z and consider the group ΣFR(Fn) ∼= FR(Fn)oSn. Using
Theorem 3.4.7 we may calculate the homology as
H∗
(
ΣFR(Fn), R
)
= H∗(Sn, R)⊕
⊕
f∈Forests
H∗
(
Aut(f), Ĉ∗(f)
)
(3.4.3.14)
∼= H∗(Sn, R)⊕
⊕
f∈Forests
t|E(f)|H∗
(
Aut(f), Rf
)
, (3.4.3.15)
where t is of degree one and Rf is the one-dimensional ‘determinant’ Aut(f)-module: a
factor of −1 is introduced every time two edges of f are swapped.
3.5 Categorical interpretation of symmetric automor-
phisms
The partial conjugations play an important role in the theory of automorphisms of free
products. In this section we aim to show that they can be viewed as generalisations of
inner automorphisms in a sense which can be made precise in the language of category
theory. In fact we may show that the Whitehead automorphism group WH(G) occurs
naturally as automorphisms which can be extended with respect to the free product
82 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
functor
∗n−1 : Gps×n → Gps. (3.5.0.16)
In comparison, the inner automorphisms of a group are those automorphisms which extend
with respect to the identity functor
IGps : Gps→ Gps. (3.5.0.17)
3.5.1 Extendable automorphisms
Let F : C → D be a functor.
Definition 3.5.1. Let C be an object of C and α : FC → FC be an automorphism of
FC in D. Then we say that the pair (C, α) is extendable with respect to F if for every
morphism f : C → B with source C there exists an automorphism β of FB making the
square
FC
α //
Ff
FC
Ff
FB
β // FB
(3.5.1.1)
commute. The extendable automorphisms {(C, α)} of FC form a group, which we denote
EFC.
Although it is not necessarily the case, in many examples the extendable automor-
phisms form a functor EF : C → Gps to the category of groups, where the commuting
squares take the form
FC
α∈EFC//
Ff
FC
Ff
FB
(EF f)(α)// FB.
(3.5.1.2)
Example 3.5.1.1. Let F be the identity functor on the category of groups, I : Gps →
Gps. Then by a theorem of Schupp [43] the extendable automorphisms consist of the
inner automorphisms.
Theorem 3.5.2 (due to Schupp [43]). Let G be a group and α an automorphism of G.
The automorphism α is an inner automorphism of G if and only if α has the property
that whenever G is embedded in a group H then α extends to some automorphism of H.
Proof. A subgroup K ≤ H is called malnormal in H if hKh−1∩K = {1} for all h ∈ H\K.
Schupp noted that it was enough to prove that any group G is embeddable as a malnormal
subgroup of a complete group H, which he then went on to prove. This is enough because
any automorphism of H is inner by the completeness of H and it restricts to G only if
the conjugating element is in G by the malnormality of the embedding.
3.5. CATEGORICAL INTERPRETATION OF SYMMETRIC AUTOMORPHISMS 83
We record Schupp’s intermediate result as the following lemma, referring back to [43]
for the proof.
Lemma 3.5.3 (due to Schupp [43]). Any group G is embeddable as a malnormal subgroup
of a complete group H.
Remark 3.5.4. The inner automorphism group is normal inside the full automorphism
group. It might be hoped that the extendable automorphism group EFC of an object FC
is normal inside the full automorphism group Aut(FC), however this is not necessarily
the case. An example will be given by the Whitehead automorphism group WH(Fn)
which is not normal inside Aut(Fn). A weaker result holds.
Proposition 3.5.5. Let F : C → D be a functor and C an object of C. Then the subgroup
F
(
Aut(C)
) ≤ Aut(FC) normalises the subgroup EFC of extendable automorphisms of F .
Proof. Let α be an extendable automorphism of FC and let γ ∈ Aut(C). We need to
show that we may complete the square
FC
FγαFγ−1//
Ff
FC
Ff
FB FB
(3.5.1.3)
for any f ∈ Hom(C,B). Redraw the above diagram as
FC
Fγ−1 //
Ff ""F
FF
FF
FF
F FC
α //
F (fγ)
FC
F (fγ)
Fγ // FC
Ff||xx
xx
xx
xx
FB FB .
(3.5.1.4)
Looking at the central square in this diagram, α may be extended along F (fγ) in order
to fill in the lower edge. The same extension serves to extend FγαFγ−1 along Ff . Hence
FγαFγ−1 is extendable with respect to F .
3.5.2 A characterisation of WH(G)
The direct product Gps×n of n copies of the category of groups has as objects n-tuples
of groups G = (G1, . . . , Gn) and the morphisms are n-tuples of group homomorphisms
f = (f1, . . . , fn). The free product functor
∗n−1 : Gps×n → Gps (3.5.2.1)
takes an n-tuple of groups and gives their free product G1 ∗ . . . ∗ Gn. So an extendable
automorphism α of the free product G1 ∗ . . . ∗ Gn with respect to the functor ∗n−1 is an
84 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
automorphism α for which each square
G1 ∗ . . . ∗Gn α //
f1∗...∗fn
G1 ∗ . . . ∗Gn
f1∗...∗fn
H1 ∗ . . . ∗Hn // H1 ∗ . . . ∗Hn
(3.5.2.2)
may be completed for each n-tuple of group morphisms
(f1, . . . , fn) : (G1, . . . , Gn)→ (H1, . . . , Hn) . (3.5.2.3)
Recall that the Whitehead automorphism group of G1 ∗ . . .∗Gn is generated by partial
conjugations α
gj
i for i 6= j and gj ∈ Gj and by inner factor automorphisms inn(gi) for
gi ∈ Gi. Both kinds of generators are extendable; indeed for (f1, . . . , fn), the generator
α
gj
i extends to α
fj(gj)
i and the generator inn(gi) extends to inn(fi(gi)). The purpose of
this section is to show the converse, that every extendable automorphism is contained in
the Whitehead automorphism group.
Theorem 3.5.6. Let G = (G1, . . . , Gn) be an n-tuple of groups and α be an automorphism
of the free product G = G1 ∗ . . . ∗ Gn. Then α is a Whitehead automorphism if and only
if it extends with respect to the free product functor ∗n−1. That is for every morphism
f1 ∗ . . . ∗ fn : G1 ∗ . . . ∗Gn → H1 ∗ . . . ∗Hn, (3.5.2.4)
there exists an automorphism β of H1∗ . . .∗Hn such that the following diagram commutes.
G1 ∗ . . . ∗Gn α //
f1∗...∗fn
G1 ∗ . . . ∗Gn
f1∗...∗fn
H1 ∗ . . . ∗Hn β // H1 ∗ . . . ∗Hn.
(3.5.2.5)
Proof. We have already seen that every Whitehead automorphism extends. For the op-
posite implication, suppose that α is an automorphism of G which satisfies the extend-
ing condition. By Lemma 3.5.3 each Gi is embeddable as a malnormal subgroup of a
complete group Hi. Since each Hi is complete, it’s true that each Hi is neither freely
decomposable nor isomorphic to Z. And so the automorphism group Aut(H1 ∗ . . . ∗Hn)
is ΣWH(H1 ∗ . . . ∗Hn).
We now show that any γ ∈ ΣWH(H1 ∗ . . . ∗ Hn) restricting to G1 ∗ . . . ∗ Gn must
restrict to an element of ΣWH(G1 ∗ . . . ∗Gn). For g ∈ Gi, the action of γ is given by
γ(g) = (g′)hi , (3.5.2.6)
where g′ = g ∈ Gj for some isomorphic factor Gj ∼= Gi and for some fixed hi ∈ H1∗. . .∗Hn.
3.6. RELATIONSHIP WITH OUTER SPACE 85
Reducing (g′)hi to a reduced word in the letters H1, . . . , Hn we get
γ(g) = ((g′)h
′
i)w, (3.5.2.7)
where h′i ∈ Hj and w is a reduced word with its first letter not in Hj. Since γ(g) ∈
G1 ∗ . . . ∗ Gn this implies that gh′i is in Gi and w is in G1 ∗ . . . ∗ Gn. And since Gi
is malnormally embedded in Hi, the element h
′
i must be in Gi, hence hi = h
′
iw is in
G1 ∗ . . . ∗Gn. Therefore γ does restrict to an an element of ΣWH(G1 ∗ . . . ∗Gn).
The automorphism α extends to an automorphism β of H1 ∗ . . . ∗ Hn, which by the
above must be in ΣWH(H1 ∗ . . . ∗Hn). But α is the restriction of β to G1 ∗ . . . ∗Gn and
so must itself by in ΣWH(G1 ∗ . . . ∗Gn).
We have proved that α is in the symmetric Whitehead group, it remains to show
that it is a pure Whitehead automorphism. We may embed the Gi into pairwise non-
isomorphic groups, Ki, none of which are free decomposable or Z. So α extends to
κ ∈ Aut(K1 ∗ . . . ∗Kn). Since no two factors Ki are isomorphic the automorphism group
of their free product is generated by factor automorphisms and partial conjugations. So
κ must take g ∈ Gi ≤ Ki to a conjugate of an element of Gi, but since κ(g) = α(g), this
must be a conjugate of g. Hence α is a pure Whitehead automorphism.
3.6 Relationship with Outer space
The moduli space of cactus products is pictured as a ‘space of spaces’ each of which has
fundamental group a free product of groups. Likewise Outer space is a ‘space of graphs’
each of which has fundamental group the free group. So it is natural to compare both
concepts to look for both similarities and differences. But it is also natural to ask if the
two concepts can be united into one space which would act as an Outer space for a general
free product in which the integer factors are taken into account.
3.6.1 Background on Outer space
Outer space was defined in [17] in order to study the automorphism group of a free group.
In that paper the authors show that Aut(Fn) and Out(Fn) have virtual cohomological
dimensions 2n− 2 and 2n− 3 respectively by showing that they act with finite stabilisers
on contractible spaces of those respective dimensions. The spaces are called Auter space
and Outer space2 and are both spaces of marked metric graphs, the difference being that
the graphs in Auter space have a basepoint.
A graph here will be a one dimensional finite CW-complex of fixed rank r. A metric
graph is such a graph where each edge is assigned a length. A graph G is called marked
2to be precise these spaces have greater dimensions but they equivariantly retract to spaces (the
spines) which do have the required dimensions
86 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
when it possesses a homotopy class of maps
m :
r∨
i=1
S1 → G. (3.6.1.1)
The marking m must represent a homotopy equivalence. The main result of [17] is that
the spaces of such graphs are contractible and that the natural action by Aut(Fn) and
Out(Fn) respectively have finite stabilisers.
3.6.2 Cactus graphs
Let Y = S1 be the circle with a chosen basepoint. Then a cactus product is a glueing of
circles and in particular is a graph. We call the graphs obtained in this manner cactus
graphs.
A cactus graph has a basepoint and at each vertex the valence is even, also every
vertex is a separating vertex. The moduli space of cactus graphs is a classifying space for
its fundamental group FR(Z∗n). Its universal cover E may be described by taking the
space of pairs, each consisting of a cactus graph G and a marking (3.6.1.1). Assigning
the usual metric to S1 we have a moduli space of marked metric pointed graphs and so E
maps into Auter space. The property of being a cactus graph is quite restrictive, indeed
the dimension of the moduli space of cactus graphs is (n− 1), half that of Auter space.
Cactus graphs without basepoint were considered by Collins in [14], although they
are referred to as symmetric graphs. Their moduli space is also contractible and this
was shown using ‘symmetric analogues’ of the techniques used by Culler and Vogtmann
in [17].
3.6.3 Outer space with boundaries
An Outer space with boundaries was studied in [29]. The boundaries are circles embedded
inside the marked metric graphs, the corresponding groups are subgroups of Aut(Fn)
which fix certain conjugacy classes. When the total rank of the graphs is contributed
solely by the boundaries we reobtain the space of symmetric graphs (or marked unpointed
cactus graphs).
A similar space, this time called relative Outer space was defined in [38]. This is similar
to Outer space with boundaries, however now the embedded circles may be replaced with
embedded wedges of circles. This time when the total rank of the graph is contributed
by the embedded wedges the space obtained may be compared to the unpointed cactus
products of wedges of circles.
Definition 3.6.1. These two constructions motivate a common generalisation of Outer
space and cactus products. Let Y = (Y1, . . . , Yn) be an n-tuple of pointed spaces. Then
3.6. RELATIONSHIP WITH OUTER SPACE 87
a Y-graph is a space A
• containing each Yi as a subspace
• with Yi ∩ Yj either empty or a single point for distinct i, j and
• where A/Y, the space given by contracting each subspace Yi to a point in turn, is
a graph.
For example if Y1 = S
2 and Y2 = S
1 × S1 then a Y-graph could look like
(3.6.3.1)
The rank of A is defined to be the rank of A/Y. The fundamental group of A is
pi1(Y1) ∗ . . . ∗ pi1(Yn) ∗ Fr, (3.6.3.2)
where r is the rank of A/Y.
A marked Y-graph (A,m) is a Y-graph with a homotopy class of maps
m :
n∨
i=1
Yi ∨
r∨
i=1
S1 → A, (3.6.3.3)
we also ask that m is a homotopy equivalence and that for each Yi, m preserves the
conjugacy class of the image of pi1(Yi).
By placing a metric on the edge components of A we may consider marked metric
Y-graphs. Define OS(Y ) to be the space of marked metric Y-graphs. This may be
topologised in the standard way, but also using the topology of the Yi.
Conjecture 3.6.1. Let Y be an n-tuple of pointed spaces and suppose that each Yi is
aspherical. Then OS(Y) is connected and contractible.
Remark 3.6.2. This moduli space of marked metric Y-graphs can be directly compared
with the Outer space of a free group. It is tempting to call this space the Outer space of
a free product. However there is already a notion of the Outer space of a free product
88 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
as introduced in [23]. The definition in that paper is a hydrid of the McCullough-Miller
space of a free product and the Outer space of a free group.
Remark 3.6.3. An approach to a proof of this conjecture is to view the Outer space of
Guirardel and Levitt as an analogue of McCullough-Miller space and to attempt to show
that the contractibility of the moduli space defined above is equivalent to the contractibil-
ity of the Outer space, which was proved in [23].
A corollary of this conjecture would be a formula for the virtual cohomological dimen-
sion of Aut(G) for any finitely generated G, in terms of the factors of the free product
decomposition of G.
3.7. AUTOMORPHISMS OF GRAPH PRODUCTS 89
3.7 Automorphisms of graph products
In this section we will use diagonal complexes to construct groups which act on graph
products of groups. Let G = (V,E) be a graph, then for each V -tuple of groups H there
is a graph product HG. We are interested in automorphisms which act by conjugation
on the vertex subgroups.
We will proceed by defining a diagonal complex ΓG which depends only on the graph
G. Next we will show that the diagonal complex products HΓG act on HG by symmetric
automorphisms. This will be followed by a discussion of the literature on automorphisms
of graph products, with the outcome that we do not know whether the group of partial
conjugations of HG is isomorphic to HΓG or not. Finally we will approach the ques-
tion of whether the diagonal complex ΓG is aspherical
3 by defining an analogue of the
McCullough-Miller complex. The asphericity of ΓG is equivalent to the contractibility of
these complexes.
3.7.1 The diagonal complex ΓG
Let w be a vertex of G and write Gw for the full subgraph of G with vertices in the
complement of the star of w. So this is the graph given by taking away from G the vertex
w and any other vertices joined by a single edge to w. We say that a subset A ⊆ V is
admissable with respect to w if it is the vertex set of a connected component of Gw. The
set of admissable sets with respect to w is denoted adm(w). This may be represented by
the diagram
(3.7.1.1)
where adm(w) = {A1, . . . , Ak}.
Definition 3.7.1. For a graph G = (V,E) let XG be the set of pairs (w,A) with w ∈ V
and A ∈ adm(w). We call such a pair (w,A) a G-admissable pair. A G-admissable poset
(V,≤) has vertex set V , is non-trivial and satisfies the following two conditions:
3Recall that a diagonal complex is aspherical if the diagonal complex products of aspherical spaces
are themselves aspherical.
90 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
1. (the ‘overset condition’) For every w ∈ V , the set {v | w < v} is the union of sets
admissable with respect to w.
2. (the ‘underset condition’) For every w ∈ V , any two elements x, y in {v | v < w}
satisfy either
• x ≤ y,
• y ≤ x or
• (x, y) ∈ E is an edge in G.
We denote by ΓG the set of G-admissable posets.
Since the overset condition implies that {(v, w) | v < w} is the union of sets (w,A) ∈
XG we may view any G-admissable poset as a subset of XG and so ΓG as a subset of PfXG.
When v < w in a G-admissable poset we write [v < w] for the element (v, A) ∈ XG where
A is the v-admissable subset containing w.
Example 3.7.1.1. Let G be the graph
1
2
3
4
5
6
sss
KKK
(3.7.1.2)
and let P be the poset
2 4
6
@@
^^====
1
@@
5
^^====
(3.7.1.3)
Then P is a G-admissable poset, it is the union of the elements (1, 24), (1, 6), (5, 24), (5, 6)
and (6, 24) of XG.
Let P ∈ ΓG and v < w in P . Choose a maximal path4
v = v0 < v1 < . . . < vr = w (3.7.1.4)
between v and w. Let A be the v-admissable subset containing v1, we claim that A does
not depend on the choice of maximal path and so we may define a map γPG : {v < w} → XG
taking v < w to (v,A). Indeed, choose another maximal path this time with v′1 the first
step, then by the underset condition applied to w the vertices v1 and v
′
1 are joined by an
edge and therefore v1 and v
′
1 are contained in the same connected component of Gv.
4a maximal path has the property that if vi < v
′ < vi+1 then v′ is equal to either vi or vi+1.
3.7. AUTOMORPHISMS OF GRAPH PRODUCTS 91
Lemma 3.7.2. Let P be a G-admissable poset.
1. Let v < w < w′ ∈ P , then γPG(v < w) = γPG(v < w′).
2. Let v be a vertex of G and let w and w′ be two vertices in the same v-admissable
subset, then γPG(v < w) = γ
P
G(v < w
′).
Proof. The concatenation of maximal paths from v to w and from w to w′ is a maximal
path from v to w′. The first step from the path from v to w is the same as the first step
of the composite path from v to w′, hence γPG(v < w) = γ
P
G(v < w
′).
Now for the second part. The vertices w and w′ are in the same connected component
of Gv and so it is enough to assume that w and w
′ are connected by an edge as equality
under γPG in this case would imply equality for the whole connected component. Let
v = v0 < v1 < . . . < vr−1 < vr = w (3.7.1.5)
and
v = v′0 < v
′
1 < . . . < v
′
s−1 < v
′
s = w
′ (3.7.1.6)
be maximal paths in P . Suppose that r = s = 1, then γPG(v < w) = [v < w] and
γPG(v < w
′) = [v < w′], which by assumption are the same class so in this case we are
done.
Now suppose that r 6= 1, then v1 < w so w is not in the star of v1. Since w′ is
connected by an edge to w either w′ is in the same connected component of Gv1 so
v < v1 < w
′ or w′ is in the star of v1 and hence is joined by an edge to v1. In the first
case γPG(v < w
′) = [v < v1] = γPG(v < w) using part 1) of the lemma. In the second case
we may replace w by v1 because γ
P
G(v < v1) = γ
P
G(v < w) and v1 is joined by an edge to
w′. Our ‘new’ r is 1 and s remains unchanged. Hence we may assume that r = 1. The
same argument applies to s meaning we may assume that s = 1 also and we have already
covered the case r = s = 1.
Definition 3.7.3. The previous lemma established that γPG is constant on subsets {v < w}
where w ranges across a v-admissable subset. Hence γPG can be lifted to P considered as
a subset of XG. We define γG to be the fibres of this lifting, explicitly this is
γG(P ) =
{
(γPG)
−1(v,A) | (v,A) ∈ Im(γPG)
}
. (3.7.1.7)
Example 3.7.1.2. Recall the graph G and G-admissable poset P from Example 3.7.1.1.
The image of γPG is
(1, 6), (5, 6) and (6, 24). (3.7.1.8)
92 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
By taking the fibres we find that
γG(P ) =
{
2 4 6
1
^^====
OO @@ ,
2 4 6
5
^^====
OO @@ ,
2 4
6
@@
^^====
}
. (3.7.1.9)
Proposition 3.7.4. The pair (ΓG, γG) is a diagonal complex.
Proof. We must show that (1)-(3) from Definition 2.1.3 hold. Throughout this proof P
will any G-admissable poset.
Condition (1) states that each (v, A) is a G-admissable poset, this is trivially true.
That γG(P ) is a partition of P is clear because γG(P ) consists of the fibres of the map
P → XG. To complete condition (2) we need only remark that γPG is constant only if
P ∈ X+G . Now onto the first part of condition (3); we must show that any union of a
subset of γG(P ) is itself a G-admissable subset. Let A ⊆ γG(P ) and let
PA =
⋃
U∈A
U (3.7.1.10)
be the union. To see that PA is a poset we need only show transitivity, let v1 < v2 and
v2 < v3 in PA. Since
γPG(v1 < v2) = γ
P
G(v1 < v3) (3.7.1.11)
we have v1 < v3 in PA. Next to see that PA has the overset condition we need only note
that by definition it is a union of elements of XG. For the underset condition suppose
that v < w and v′ < w in PA. Since PA ⊆ P the underset condition for P implies that
either v ≤ v′, v′ ≤ v in P or (v, v′) is an edge of G. In the case that v < v′ we have that
γPG(v < v
′) = γPG(v < w) so v < v
′ in PA. Thus the underset condition holds for PA.
Now it only remains to show that γG(PA) is a partial refinement of γG(P ). This follows
from
γPG |PA = γPG ◦ γPAG (3.7.1.12)
which implies that the fibres of γPAG are a partial refinement of the fibres of γ
P
G . To
show (3.7.1.12), let v < w in PA and let
v = v0 < v1 < . . . < vr−1 < vr = w (3.7.1.13)
be a maximal path from v to w in PA, then γ
PA
G (v < w) = [v < v1]. But v < v1 < w in P
so γPG(v < w) = γ
P
G(v < v1), which establishes (3.7.1.12).
Example 3.7.1.3. Still examining the graph G and G-admissable poset P from Exam-
3.7. AUTOMORPHISMS OF GRAPH PRODUCTS 93
ple 3.7.1.1 we may give the maximal faces of P as
2 4 6
1
^^====
@@
55kkkkkkkkkkkkk 5
iiTTTTTTTTTTTTT
^^====
@@ ,
2 4
6
@@
^^====
1
OO and
2 4
6
@@
^^====
5
OO . (3.7.1.14)
Proposition 3.7.5. The diagonal complex (ΓG, γG) is proper. In addition with the map
l : XG → V given by l(v, A) = v it is a V -labelled diagonal complex.
Proof. First that the diagonal complex is proper. Let P be a G-admissable poset and let
(v,A) ∈ γPG(P ). It is enough to show that
P ′ = P − (γPG)−1(v, A) (3.7.1.15)
is a maximal proper G-admissable poset inside P . Let P ′ ⊂ P ′′ ⊆ P be an intermediate
element of ΓG and suppose that v < w in P
′′ but not in P ′. Let v1 be the first step of a
path from v to w inside P , then v1 ∈ A. Now inside P ′′ we have v, v1 ≤ w and so by the
underset condition either v ≤ v1, v1 ≤ v or (v, v1) is an edge, but since P ′′ is contained in
P it must be that v ≤ v1. Hence (v, A) = [v < v1] ∈ P ′′. For any v < w′′ in (γPG)−1(v,A)
we may find an intermediate w′ ∈ A such that v < w′ < w′′ in P , since w < w′′is in P ′ and
v < w′ in P ′′ this implies that v < w′′ in P ′′ and hence P = P ′′. Hence the maximal faces
of P are the maximal G-admissable posets in the inclusion order, so (ΓG, γG) is proper.
That ΓG is V -labelled is a simple observation; any U ∈ γG(P ) is a fibre of the map
γPG :
(γPG)
−1(v, A) (3.7.1.16)
which consists of pairs (v,B) hence l is constant on elements of γG(P ).
3.7.2 Action on the graph products
The automorphism groups of right-angled Artin groups have similar properties and a
similar ‘flavour’ to the automorphism groups of the free groups. Ofcourse a right-angled
Artin group based on a discrete graph is a free group. At the other extreme the RAAG
associated to a complete graph is a free abelian group and the automorphism group
is GLn(Z). So the automorphism groups of RAAGs may be considered to interpolate
between Aut(Fn) and GLn(Z). The relations of a right-angled Artin group provide re-
strictions to its possible automorphisms, however the automorphism groups are still large
and the added complexity of the relations means that the automorphism groups are more
complicated and trickier to study. In particular the presentation of the automorphism
group depends on the graph.
The majority of work on automorphism groups of graph products has focussed on
94 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
the right-angled Artin group case, see for instance [19] and [12]. However there are also
treatments of more general cases, for graph products of abelian groups see [24] and for
results for a restricted class of graphs see [11].
Let G = (V,E) be a graph and H be a V -tuple of groups. Recall from Chapter 1 that
HG is the group with presentation ∐
v∈V Hv
〈[Hv, Hw] | vw ∈ E〉 . (3.7.2.1)
Let (v, A) be a G-admissable pair and h ∈ Hv. Then αh(v,A) is an element of HΓG and by
Theorem 2.2.1 the set of all such αh(v,A) generates HΓG. The action of α
h
(v,A) on HG is
defined as follows
αh(v,A)(g) =
gh if g ∈ Hw for w ∈ A,g if g ∈ Hw for w /∈ A. (3.7.2.2)
To check that this action of αh(v,A) on HG is well-defined we must check that α
h
(v,A) preserves
the relation of HG; so let e ∈ E be an edge joining w and w′ in G, then for gw ∈ Hw and
gw′ ∈ Hw′ there is a relation
[gw, gw′ ] . (3.7.2.3)
Suppose that both w,w′ ∈ A, then
αh(v,A)([gw, gw′ ]) =
[
ghw, g
h
w′
]
= [gw, gw′ ]
h . (3.7.2.4)
For w,w′ /∈ A, then
αh(v,A)([gw, gw′ ]) = [gw, gw′ ] . (3.7.2.5)
Now without loss of generality we are left with the case that w ∈ A and w′ /∈ A. Since
(v,A) is an admissable pair, the set A is a connected component of Gv = G− st(v). But
e is an edge with one end w in A and the other end w′ not in A, therefore w′ must be in
the star of v, hence gw′ commutes with h and in particular gw′ = g
h
w′ . Therefore
αh(v,A)([gw, gw′ ]) =
[
ghw, gw′
]
=
[
ghw, g
h
w′
]
= [gw, gw′ ]
h . (3.7.2.6)
This concludes the check that each αh(v,A) defines an automorphism of HG, we will denote
this automorphism βh(v,A) to distinguish it from the element of HΓG. Our next task is
to check that the relations of HΓG coming from the diagonal complex ΓG hold for the
automorphisms βh(v,A). Relations for ΓG come from the two dimensional elements, first
3.7. AUTOMORPHISMS OF GRAPH PRODUCTS 95
there are the elements of the form {(v,A), (w,B)}, the associated posets take the forms
A B
v
<<<<<
w
=====
, (3.7.2.7)
A ∪B
v
:::::
ttttt
jjjjjjjjjjjjjj w
JJJJJ
TTTTTTTTTTTTTT
, (3.7.2.8)
A B
v = w
UUUUUUUUUUUUU
QQQQQQQQ
GGGGGG
wwwwww
mmmmmmmm
iiiiiiiiiiiii , (3.7.2.9)
B
w
=====
v
>>>>>
. (3.7.2.10)
In the final tree (3.7.2.10) there is the inclusion {w} ∪B ⊆ A. The other elements are of
the form
{(v, A), (w,B), (v, C1), . . . , (v, Ck)}, (3.7.2.11)
where
⋃
i=1,...,k Ci ⊆ B, w ∈ A and B ⊆ A∪
⋃
i=1,...,k Ci. This forms a tree of height three
just as in (3.7.2.10). The relations for (3.7.2.7)-(3.7.2.10) are of the form
[
βh(v,A), β
g
(w,B)
]
,
whereas the relation for (3.7.2.11) is[
βg(w,B), β
h
(v,A)β
h
(v,C1)
. . . βh(v,Ck)
]
. (3.7.2.12)
This final relation may be simplified; the elements βh(v,A), β
h
(v,C1)
, . . . , βh(v,Ck) commute by
the relation (3.7.2.9) and if we let A′ = A ∪ C1 ∪ . . . ∪ Ck be their union and write
βh(v,A′) = β
h
(v,A)β
h
(v,C1)
. . . βh(v,Ck), (3.7.2.13)
then (3.7.2.12) becomes
[
βg(w,B), β
h
(v,A′)
]
and w ∪ B ⊆ A′ and so this relation may be
grouped with (3.7.2.10).
Checking these relations is little more complicated than for the automorphisms of free
products; for (3.7.2.7) and (3.7.2.9), the sets A and B are disjoint and so the associated
automorphisms act non-trivially on different subsets of the generators and hence commute.
For the tree (3.7.2.8) the sets A and B do overlap, however by the underset condition v
and w must be joined by an edge in G and so in H the elements g and h commute, so the
relation holds.
We are left with the case that w ∪ B ⊆ A, where A may be a union of v-admissable
96 3. SYMMETRIC AUTOMORPHISMS OF FREE PRODUCTS
sets. Let w′ ∈ B and g′ ∈ Hw′ then
βg(w,B)β
h
(v,A)(g
′) = βg(w,B)(g
′h) = (g′g)h = g′gh and (3.7.2.14)
βh(v,A)β
g
(w,B)(g
′) = βg(w,B)(g
′g) = (g′h)g
h
= g′gh. (3.7.2.15)
The remaining cases where w′ /∈ B are simpler.
We have just shown the following.
Proposition 3.7.6. There is a map
HΓG → Aut(HG). (3.7.2.16)
Remark 3.7.7. The functoriality of HΓG in H means that those automorphisms are ex-
tendable in the sense of Section 3.5.1.
Question A. Is the action of HΓG on HG is faithful? Also do all partial conjugations
arise in this way?
To answer this question we would need different methods, perhaps peak reduction,
which we have not covered in this thesis. The next question to ask is
Question B. For any graph G, is the diagonal complex ΓG aspherical?
This question is a direct analogue of Theorem 3.3.12 and in fact when G is the discrete
graph reduces to the theorem. The most direct method of investigating this question
appears to by using the associated coset complex for each n-tuple of groups G. This
coset complex may be seen as a candidate for an analogue of McCullough-Miller space
and it suffices to prove that it is contractible.
A positive answer to both of these questions would allow one to compute the homology
of the symmetric automorphism group of a graph product. The answer would be written
in terms of direct sums indexed by G-graphs.
Chapter 4
Configuration spaces
Much of the initial interest in symmetric automorphism groups was concentrated on the
symmetric automorphism group of a free group, FR(Z∗k) and one of the main motivations
for studying this group was a result contained in the thesis of Dahm [18]. He showed that
the fundamental group of a configuration space of k oriented loops with an unknotted,
unlinked embedding in R3 was FR(Z∗k).
These spaces are of interest in physics, for instance the equations which govern the
dynamics of an inviscid fluid in three dimensions force any vorticity (the curl of the velocity
field) to lie along embedded paths, elsewhere it is zero. Furthermore the vorticity is a
constant of any solution of the equations; when the fluid moves the vorticity moves with
it and furthermore its strength at that point is conserved. The strength of the vorticity
is constant along the line and the line must either be infinite or it joins up with itself to
form a vortex ring. As such solutions to these dynamical equations have a topological
order.
These results are due to Hermann von Helmholtz [26]. William Thomson (later to
become Lord Kelvin) was so inspired that he championed [30] a “Vortex theory of Atoms”
suggesting that ring vortices occuring in the ‘ether’ could account for the fundamental
particles of nature. The motivation for this idea was the permanence of the vortex rings,
they could neither be created nor destroyed, and also by being knotted or linked they
could provide enough variety to account for all possible particles. However the theory
made no successful predictions and was subsequently abandoned. Furthermore Kelvin was
frustrated by the fact that he could not construct stable solutions of the hydrodynamical
equations except in the simplest of examples. The theory of vortex rings is still an active
area of study today, for example [2] and [31]. For a good summary of the history and
current state of vortex dynamics see [40].
There has been a large overlap between work on FR(Z∗k) and on these configuration
spaces. Relevant works include the thesis of Dahm [18] and papers [36], [14], [9], [5]
and [27]. The roles of these papers were covered in the introduction to the thesis. Results
of particular importance to configuration spaces of loops include the result of Dahm that
97
98 4. CONFIGURATION SPACES
the fundamental groups correspond to the groups of partial conjugations FR(Z∗k) of
the free group, in [42] there is discussion of fibrations involving these spaces and more
recently in [6] it is proved that the configuration space of smooth embeddings is homotopy
equivalent to a space of Euclidean embeddings.
It is the Euclidean embeddings which we consider in this chapter, however the particu-
lar example of configuration spaces of loops is extended to a family Ln(k) of configuration
spaces of k copies of the n-sphere embedded into Rn+2, we assume that n ≥ 1. Since we
consider Euclidean embeddings the configuration spaces are manifolds and our first task is
to describe the manifold strata associated to them. This allows us to prove Theorem 4.1.3
which states that the fundamental group pi1(Ln(k)) is FR(Z∗k) and so is independent of
n. So in terms of the fundamental groups we find the same result for n-spheres as we do
for loops. The proof is a direct translation from Proposition 3.4 of [6].
The next section considers other homotopy invariants of the spaces Ln(k); we con-
sider the integral homology, the higher homotopy groups and the homotopy type of the
suspension. These are significantly more challenging to compute however conjectures are
made; these include a precise characterisation of the integral homology and a connectivity
result concerning the higher homotopy groups. We show that the first conjecture would
imply that the suspension of Ln(k) is homotopy equivalent to a bouquet of spheres. For
k = 2 direct calculations are simple and so the conjectures are checked for this value.
We finish the section by considering the colimit of the spaces Ln(k) as n ranges over N.
This is denoted L∞(k) and it would follow from the connectivity conjecture that it is a
classifying space for FR(Z∗k). The advantage over the classifying space constructed in
the previous chapter is that the action of Sk on L∞(k) given by permuting the embedded
spheres is proper and so the quotient by Sk is a classifying space for
ΣFR(Z∗k) ∼= FR(Z∗k)oSk. (4.0.2.1)
We finish the chapter by sketching a possible proof of the homology conjecture.
4.1 The spaces of codimension 2 spheres
There are a number of different ways to produce a space of embeddings of n-spheres into
Rn+2, one could take the space of smooth embeddings, or spaces of Euclidean embed-
dings. These are not necessarily equivalent and our choice of Euclidean embeddings via
translations and scalings is chosen to give a space with elegant behaviour but without too
complicated geometry.
4.1. THE SPACES OF CODIMENSION 2 SPHERES 99
4.1.1 Some definitions
Let Sn ⊆ Rn+2 be the n-sphere with the following embedding{
(x1, . . . , xn+1, 0) ∈ Rn+2 |
n+1∑
i=1
x2i = 1
}
. (4.1.1.1)
For example in R3, the 1-sphere S1 is the unit circle lying in the xy-plane with centre the
origin. The n-sphere is of codimension 2, nevertheless we refer to Dn+1{
(x1, . . . , xn+1, 0) ∈ Rn+2 |
n+1∑
i=1
x2i < 1
}
. (4.1.1.2)
as the interior of Sn. Let TSn+2 = Rn+2oR>0 be the group of translations and scalings
acting on Rn+2, so
(y1, . . . , yn+2, t) (x1, . . . , xn+2) = (y1 + tx1, . . . , yn+2 + txn+2) . (4.1.1.3)
Then the stabiliser StabTSn+2(S
n) = {e} and so we identify the space of embeddings of the
n-sphere with TSn+2. The sphere s associated to an element g = (y1, . . . , yn+2, t) ∈ TSn+2
is
g.Sn =
{
(x1, . . . , xn+1, yn+2) |
n+1∑
i=1
(xi − yi)2 = t2
}
. (4.1.1.4)
Let g = (g1, . . . , gk) ∈ TSkn+2, then we write si for the sphere gi.Sn. We say that si and
sj are disjoint if si ∩ sj = ∅.
Definition 4.1.1. Let Ln(k) be the subspace of TSkn+2 consisting of the g with pairwise
disjoint si. We call this the space of n-spheres in Rn+2, points of Ln(k) will be called
configurations.
The support of a configuration
Let g ∈ Ln(k) be a configuration, we define a partial ordering ≤ on [k] = {1, . . . , k}, by
writing si ≤ sj if gjDn+1 ⊆ giDn+1, that is the interior of si contains the interior of sj.
We call ([k] ,≤) the support, supp(g) of g.
Recall that a forest poset F is a non-empty poset which satisfies the underset condi-
tion (3.3.1.1):
for all x ∈ F, {y | y ≤ x} is a total order. (4.1.1.5)
Lemma 4.1.2. Let g ∈ Ln(k), then the support supp(g) is either the empty poset or a
forest poset.
Proof. Let l ∈ [k] be a point of supp(g) and suppose that si < sl and sj < sl. Then both
of the interiors of si and sj contain the interior of sl and hence the interiors of si and
100 4. CONFIGURATION SPACES
sj have non-empty intersection. But this implies that one must be contained within the
other, hence either si ≤ sj or sj ≤ si.
4.1.2 The structure of the configuration space
Recall that by viewing a poset as a subset of {(i, j) ∈ [k]× [k] | i 6= j}, we may form the
poset of posets on [k] ordered under inclusion. Let F be a forest poset and LFn be the
subspace of Ln(k) consisting of those configurations g with supp(g) ⊆ F . Let L0n be the
subspace of Ln(k) of configurations with empty support. We denote by L(F )n the subspace
of configurations with support equal to F , hence as sets
LFn = L0n q
∐
F ′⊆F
L(F ′)n . (4.1.2.1)
Of course
Ln(k) = L0n q
∐
F
L(F )n . (4.1.2.2)
Let F be a forest poset and let j1, . . . , jr be the roots of the trees within the Hasse diagram
of F , equivalently these are the minimal elements in F . Let g ∈ L(F )n be a configuration
with support F . Since each js is minimal the sphere sjs can not be contained in any other
spheres. Also if i 6= js for any s = 1, . . . , r, then si must be contained in one of the sjs .
So there is a projection
P : L(F )n → Xrn+2 (4.1.2.3)
where Xrn+2 is the space of disjoint embeddings (via elements of TSn+2) of r copies of
Dn+1 into Rn+2. This space is homotopic to the configuration space Confr(Rn+2) of r
points in Rn+2, which can be seen by sending the discs Dn+1 to their centre points. The
map going the other way is given by sending each point to the disc with centre that point
and radius a quarter of the minimal distance between points, m say, the homotopy is then
given by first shrinking the discs with radius larger than m and then enlarging those discs
with smaller radii.
Now consider any element i of F , suppose that i has t outgoing edges. This means
that for each configuration the sphere si contains within it t other spheres corresponding
to the other ends of the outgoing edges. So let Y tn+1 be the space of t disjoint Euclidean
n+ 1 discs embedded into a larger n+ 1 disc. Then there is a projection
pi : L(F )n → Y tn+1 (4.1.2.4)
which sends the configuration g to the configuration of t spheres embedded in the sphere
si. Using the same argument as above, the space Y
t
n+1 is homotopic to the configuration
space of t distinct points in Dn+1 and since Dn+1 is open and hence homeomorphic to
4.1. THE SPACES OF CODIMENSION 2 SPHERES 101
Rn+1, the space Y tn+1 is homotopic to Conft(Rn+1).
The projections P and pi determine a homeomorphism
P ×
∏
i∈[k]
pi : L(F )n → Xno. of treesn+2 ×
∏
i∈[k]
Y
|out(i)|
n+1 . (4.1.2.5)
Since each Y tn+1 is n-connected and each X
r
n+2 is (n + 1)-connected we now know that
L(F )n is n-connected. Applying the above arguments to L0n we find that it is homotopic to
Confk(Rn+2) and so is (n+ 1)-connected.
The manifold strata
We will now show that there is a good stratification of Ln(k) which will allow us to use
general position arguments in the proof of Theorem 4.1.3.
The space of n-spheres in Rn+2 is an open subset of TSkn+2 ∼= Rk(n+3), making it a
k(n+ 3)-manifold. Using (4.1.2.5) we may see that for a forest poset F , the space L(F )n is
of dimension
(n+ 3)(no. of minimal elements) + (n+ 2)
∑
i∈[k]
(no. of outgoing edges from i), (4.1.2.6)
which by some simple combinatorics is k(n+ 3)−|E(F )|, so L(F )n is a codimension |E(F )|
submanifold of Ln(k). In particular Ln(k) is the union of L0n and strictly positive codi-
mension strata, hence L0n is dense and open in Ln(k). We define L1n to be the union of
the codimension 1 strata
L1n =
⋃
(i,j)
L−→ijn (4.1.2.7)
where L−→ijn is L(F )n for F the forest with a single edge
−→
ij . Also we write L2n for the
codimension 2 strata
L2n =
⋃
i,j,k,l distinct
L(−→ji ,
−→
lk)
n ∪
⋃
i,j,k distinct
L(
−→
ki,
−→
kj)
n ∪
⋃
i,j,k distinct
L(
−→
kj,
−→
ji)
n . (4.1.2.8)
Here the respective terms correspond to L(F )n for forests F
i k
j
OO
l,
OO i j
k
OO AA and
i
j
OO
k
OO (4.1.2.9)
respectively.
102 4. CONFIGURATION SPACES
4.1.3 The fundamental group
The following is a straight generalisation of Proposition 3.4 of [6].
Theorem 4.1.3. Let n ≥ 1, then the fundamental group of Ln(k) is isomorphic to
FR(Z∗n).
Proof. We proceed via a general position argument. Any path in Ln(k) can be moved
infinitesimally so that it passes through the submanifolds L(F )n transversally. This means
that any path γ : S1 → Ln(k) may be perturbed to some smooth γ̂ so that it lies in L0n
and the codimension 1 strata L1n. Hence
pi1(L0n ∪ L1n)→ pi1(Ln(k)) (4.1.3.1)
is surjective. Furthermore the transversality implies that for any p ∈ S1 with γ̂(p) ∈ L1n
there is a neighbourhood U such that (γ̂ |U)−1 (L1n) = {p}. Since both L0n and each L
−→
ij
n
are simply connected and since L−→ijn has trivial normal bundle we have that pi1(L0n∪L1n) is
freely generated by the paths αij which pass only once through L1n in the component L
−→
ij
n .
Now suppose that H : [0, 1]2 → Ln(k) is a homotopy between transverse paths γ1, γ2.
Since [0, 1]2 is a 2 dimensional manifold with boundary we may perturb it to Ĥ so that it
lies transverse to any submanifold LFn . This means that the homotopy lies in L0n∪L1n∪L2n
and hence
pi1
(L0n ∪ L1n ∪ L2n)→ pi1(Ln(k)) (4.1.3.2)
is an isomorphism. The transverse homotopy Ĥ intersects L2n locally in isolated points.
Since each L(F )n is connected to find the relations for pi1(Ln(k)) it suffices to look at the
possible behaviour near each codimension 2 component. So we look case by case at the
forests from (4.1.2.9). The first forest F consists of two disjoint edges and so a point
of L(F )n is a configuration of spheres g in which sphere i is contained in sphere j and
sphere k is contained in sphere l and there are no other nested spheres. Such a point p is
represented by the centre point in the diagram (4.1.3.3) below
(4.1.3.3)
4.1. THE SPACES OF CODIMENSION 2 SPHERES 103
The remainder of the diagram represents the structure of a local intersection with the
homotopy Ĥ. The four quadrants are areas intersecting with L0n and the figure in each
quadrant represents a generic point in a neighbourhood of p. The four lines are labelled
by the codimension 1 strata they are contained in. The homotopy asserts that the two
possible paths travelling around the central point are homotopic and so we obtain the
relation
αji .α
l
k = α
l
k.α
j
i . (4.1.3.4)
The next forest consists of a single tree with a root k and two leaves with end vertices i
and j and a point p of L(F )n is a configuration in which spheres i and j are contained in
sphere k. The next diagram (4.1.3.5) is analogous to diagram (4.1.3.3) above.
(4.1.3.5)
This time it represents the relation
αki .α
k
j = α
k
j .α
k
i . (4.1.3.6)
The final case is a forest of one two-edged tree with root k which is joined to j which is in
turn joined to i. The point p this time is a configuration where sphere k contains sphere
j which in turn contains sphere i. The intersection with the homotopy this time is more
complex as there are six types of configurations in L0n occuring, each corresponding to an
104 4. CONFIGURATION SPACES
ordering of i, j and k as pictured in (4.1.3.7).
(4.1.3.7)
The corresponding relation thus has three generators on each side and is given by
αji .α
k
i .α
k
j = α
k
j .α
k
i .α
j
i . (4.1.3.8)
These three classes of relations along with the generators αji give a defining presentation
of pi1(Ln(k)) and this is the same as the defining presentation for FR(Z∗n), see Proposi-
tion 3.1.2.
4.2 Other homotopy invariants
Now that we know that the fundamental group of each of the spaces Ln(k) is FR(Z∗k)
we should consider other homotopy invariants of Ln(k). There are the homology groups
Hi(Ln(k),Z), the higher homotopy groups pii(Ln(k)) for i ≥ 2 and also the homotopy type
of the suspension ΣLn(k). It is easy to see that the higher homotopy groups are non-
trivial and that the homology groups differ from those of FR(Z∗k); consider the following
diagram to see a 2-sphere in L1(2).
(4.2.0.9)
We start this section with a conjecture which states precisely the homology of Ln(k);
whereas the homology of FR(G1 ∗ . . . ∗ Gn) is given by forests with edges labelled by
4.2. OTHER HOMOTOPY INVARIANTS 105
homology classes from H∗(Gi,Z), the ‘homology conjecture’ states that the homology of
Ln(k) is given by the same forests but now with vertex labelling, this time by elements
from the homology of the configuration space of points in Rn+2. Next we discuss the
suspension of Ln(k) and assuming that the homology conjecture holds we show that
ΣLn(k) is homotopic to a bouquet of spheres. This should come as no surprise because
it is known [27] that the suspension of a K(FR(Z∗k), 1) is a bouquet of spheres, as is the
suspension of the configuration space of points in Rn+2.
We then make another conjecture that Ln(k) is highly connected in the sense that
pii(Ln(k)) = 0 for 1 < i ≤ n; we refer to this as the ‘homotopy conjecture’. Our final
result shows that the colimit L∞(k) of the Ln(k) as n tends to infinity is a K(FR(Z∗k), 1)
with a proper Sk-action, we show that this follows by either the homology or homotopy
conjecture.
4.2.1 The homology groups of Ln(k)
We know the homology of FR(Z∗k) from Theorem 3.4.2; recall that Hi(FR(Z∗k),Z) is
the free graded Z-module on a basis of vertex labelled forests with i edges. By filling in
higher spheres there is a map
Ln(k)→ X, (4.2.1.1)
where X is a K(FR(Z∗k), 1) so there is necessarily a map on homology
H∗(Ln(k),Z)→ H∗(X,Z) = H∗(FR(Z∗k),Z). (4.2.1.2)
It is likely that this map is both split on homology and that the homology of Ln(k) splits
over a direct sum; ⊕
F∈Fk
, (4.2.1.3)
where Fk is the set of forest posets, just as the homology of FR(G1 ∗ . . . ∗Gk) does. I will
actually go further and give a conjectural precise description of the integral homology of
Ln(k). For this we will need some notation; the homology of the configuration space of
k labelled points in Rn is a free Z-module which I denote Cn(k). A combinatorial basis
involving terms like
1. [2, 4] . [5, [3, 6]] (4.2.1.4)
is possible but we will not go into details of that here. Let P be a forest poset with Hasse
diagram F , this diagram is a rooted forest, for example
1 2
8 6
^^====
@@
5
4
^^====
OO
3
OO
7.
(4.2.1.5)
106 4. CONFIGURATION SPACES
We write out(i) for the number of outgoing edges from the vertex i and define out(0)
to be the number of minimal elements of P or equivalently the number of roots of F . For
instance in the forest above, out(6) = out(4) = 2 and out(0) = 3.
To each F associate the graded Z-module
Cn(F ) := Cn+2
(
out(0)
)⊗ ⊗
i=1,...,k
Cn+1
(
out(i)
)
(4.2.1.6)
We may now state the conjecture.
Conjecture 4.2.1. The homology of Ln(k) is given by⊕
F∈Fk
Cn(F ) [|E(F )|] . (4.2.1.7)
The notation [|E(F )|] means that the graded Z-module is suspended by the number
of edges of a forest F .
4.2.2 The configuration space when k = 2
For this low value it is possible to compute the homotopy type using elementary methods.
For a point (g1,g2) ∈ Ln(2) let z1 and z2 be the respective (n + 2)th coordinates. Let
A = {(g1,g2) | z1 ≤ z2} and B = {(g1,g2) | z1 ≥ z2}, then Ln(2) = A ∪ B and the
intersection A ∩B is the space
{(g1,g2) | z1 = z2}. (4.2.2.1)
Both A and B are contractable so we see that Ln(2) is equivalent to the suspension of
A ∩ B. This intersection is the disjoint union of three connected components. These are
determined by their support; either s1 ≤ s2, s2 ≤ s1 or the support poset is empty. Both
the components with non-trivial support are contractable. The remaining component
is equivalent to the configuration space of two points in Rn+1 and so equivalent to an
n-sphere. So
Hi(A ∩B,Z) =
0 if n < i,
Z if i = n,
0 if 0 < i < n and
Z3 if i = 0.
(4.2.2.2)
4.2. OTHER HOMOTOPY INVARIANTS 107
The space Ln(2) is equivalent to the suspension so
Hi(Ln(2),Z) =
0 if n+ 1 < i,
Z if i = n+ 1,
0 if 1 < i < n+ 1,
Z2 if i = 1 and
Z if i = 0.
(4.2.2.3)
This agrees with the conjecture. The two components in degree 1 correspond to the forests
with a single edge, whilst the sum Z⊕Z[n+ 1] corresponds to the term of the forest with
no edges.
With this realisation as a suspension it is easy to compute the homotopy type of Ln(2):
it is a bouquet of spheres of dimensions 1, 1 and n+ 1. Of course we know that for higher
values of k that Ln(k) is not a bouquet of spheres, afterall the fundamental groups are
not free.
The projection p
We may use what we have learnt from Ln(2) to learn more about the general case Ln(k).
For a pair 1 ≤ i < j ≤ k, define the projection
pij : Ln(k)→ Ln(2) (4.2.2.4)
to be the map which forgets each sphere except for spheres si and sj. The full projection
p is given by applying each map pij to get
p : Ln(k)→
∏
1≤i 2.
108 4. CONFIGURATION SPACES
In fact we will show that it is a corollary of the homology conjecture 4.2.1. We will use
the map p defined above to show this; first we deal with the target of p. Since Ln(2)
is a bouquet of spheres, and since the suspension of a direct product of spheres is also
a bouquet of spheres (the most familiar example being the suspension of a torus), the
suspension
Σ
( ∏
1≤i0 so
we will contract further to the (k−1)-disc
{
(z1, . . . , zk) | z1 + . . .+ zk = 0 and z21 + . . .+ z2k ≤ 1
}
. (4.2.6.3)
4.2. OTHER HOMOTOPY INVARIANTS 111
What we have now is a kind of polyhedron. We may take its dual. This now has only one
(k−1)-cell which is the whole of the (k−1)-disc and is the dual of the space Ac, where
c : [k] → [1] is the constant map. This dual complex is in fact a filled-in version of the
Coxeter complex of type Ak−1. The action of Sk corresponds to the action of Sk on Rk.
There is a geometric interpretation of the dual. Define a function m on Rk by
m(z1, . . . , zk) = 1−
∑
1≤i0, .) by translation and scaling
respectively apply to Ln(k) and the map pi respects them. Furthermore the Morse function
m is also defined on Ln(k). However the stationary points are not isolated, instead we have
stationary sets again parametrised by the functions f : [k] → [k − i]. It is now possible
to argue that the Morse function decomposes Ln(k) into descending ‘cells’, however the
‘cells’ are not i-cells, they are the direct products of i-cells with the stationary sets. There
is a further complication in that the descending sets do not totally cover Ln(k), however
this may be resolved because the Morse function gives an explicit deformation retract
onto the space which is covered.
We obtain a Morse complex MC∗(Ln(k); f) with the form⊕
i=1,...,k−1
( ⊕
f :[k]→[k−i]
C∗(f)
)
[i], (4.2.6.6)
where C∗(f) is the chain complex of the stationary set corresponding to f . The stationary
sets are easy to describe. Let NLn(l) be the configuration space of n-spheres embedded
in Rn+1. Since the spheres may be nested (hence the N ), this space is not connected.
112 4. CONFIGURATION SPACES
View f : [k]→ [k − i] as an ordered partition, for each set f−1(a) there is NLn(f−1(a)),
the configuration space of n-spheres labelled by the elements of f−1(a) inside Rn+1. The
stationary set is the product of these configuration spaces:∏
a∈[k−i]
NLn(f−1(a)) (4.2.6.7)
Using the ideas from Section 4.1.2 one may see that this space is homotopy equivalent to
a disjoint union of products of configuration spaces of points in Rn+1. The disjoint union
is parametrised by forests and the homology of configuration spaces of labelled points is
free as a Z-module so we may compute the complexes C∗(f) explicitly.
We are now left with the differential of MC∗(Ln(k), f) to compute. These are induced
from the maps
NLn(AqB)→ NLn(A)×NLn(B), (4.2.6.8)
which split the configurations of spheres into two. Looking at the projection
NLn(AqB)→ NLn(A), (4.2.6.9)
the map is defined by forgetting the positions of the spheres labelled by points in B. The
differential on C∗(f) is the sum of such maps over all refinements of f .
The hardest step in the “proof” is the last, one must compute the homology of the
complex MC∗(Ln(k); f). Fortunately there is a theory of Gro¨bner basis for operads [20]
which gives an explicit basis for the homology of the configuration space of points in Rn+1.
This allows us to construct explicit maps linking the Morse complex and the homology
complex from the conjecture. The precise detail of this homotopy equivalence is the final
piece of the proof.
Epilogue
This thesis contains results on and related to families of groups occuring as automorphism
groups of free products. The exposition aims to prove these results in an efficient but most
of all clear manner. However the methods for constructing the spaces and proving the
results have been arrived at after a number of iterations. In this epilogue the author
would like to discuss some of the ideas which lie behind the thesis. Just because they are
no longer necessary does not mean that they are no longer relevant, indeed some of them
will be returned to in future work.
The first section will discuss some algebraic structure which led the author to first
investigate the automorphism groups of free products. The second section goes on to
discuss the influence of the functoriality of the Fouxe-Rabinovitch groups. It is the func-
toriality which is the inspiration behind the final section which sketches a theory which
aims to extend the objects of Chapter 4 to a whole functor of similar objects.
Additional algebraic structure
The author was first introduced to the groups FR(Z∗n) by Vladimir Dotsenko, who had
a suspicion that they may possess some additional algebraic structure. He thought this
because the Hilbert-Poincare´ series of the homology groups FR(Z∗n) behaves as if such
structure were present. Before a more detailed explanation is offered we will discuss some
more familiar groups.
Let g be a pure braid on n strands. The group of all such braids is denoted PBn
and there are many accounts of the theory of braids, see for instance [33]. Now let h
be a second pure braid this time on m strands. Choosing the ith strand of g there is a
composition g ◦i h defined by replacing strand i of g with h; a way to imagine this is to
replace strand i with a pipe, then the composition is given by simply threading the pure
braid h through the pipe before removing the pipe entirely. This composition is a group
morphism ◦i : PBn × PBm → PBn+m−1. The family of groups PBn for n = 2, 3, . . .
along with the compositions just described may be viewed as one algebraic object, a
non-symmetric operad. We will not discuss the theory of operads any further, the reader
only needs to know that they are pleasant algebraic objects with a rich theory and with
plenty of interesting examples (at least in the categories of sets and vector spaces). A
113
114 EPILOGUE
good reference is [34].
And so it turns out that the groups FR(Z∗n) also possess an operad structure. This
may seem to be a frivolous result only of interest to those who enjoy playing with exotic
algebraic objects, however it was this that first allowed the moduli space of cactus products
to be constructed. By finding a presentation for the homology of the groups as an operad
and then interpreting this presentation in the category of pointed spaces the classifying
spaces are revealed. It is somewhat unfortunate that this approach does not help one to
prove that the relevant spaces actually are classifying spaces, to do so it is simplest to
strip out all reference to operads and use other methods to describe the space. However
the details of the operadic picture will be communicated in a future publication.
The importance of functors
The fact that the groups FR(G1 ∗ . . . ∗ Gn) are functorial in the factor groups was not
directly used in the proof of any main theorem. However it is an observation whose
importance should not be underestimated. One reason is that it is usually much easier
to prove a theorem for a family of groups indexed by a functor because the possible
techniques are constrained to those that can be made functorial.
In this instance the author was originally interested in the moduli space of cactus
graphs and the obvious method to prove that this space is aspherical is to prove that it
is locally CAT(0). However it is not locally CAT(0) as a check on the link will show; one
may try other methods inspired by geometry to try to show that this space is aspherical
and I’m sure that it’s possible that one may work. On the other hand if presented with
the whole functor of moduli spaces of cactus products one must give up on this approach
immediately. Afterall the geometry would have to cope with all manner of exotic spaces
because the properties of pointed spaces are reflected in the image of the functor MY.
This is a typical example of a situation where trying to prove a statement for a class of
examples is easier than trying to prove it for just one example.
It is the observations of Section 3.5 that are most intriguing. They suggest that the
Whitehead automorphisms of a free product are special in the same sense that the inner
automorphisms of a general group are special. An analogous result for the moduli space
of cactus graphs is conspicuously missing, perhaps the correct statement for this would
give a completely new proof that the moduli spaces give classifying spaces. The correct
language and formalism most likely lies in the emerging field of higher category theory.
The exciting prospect is that a general theory could give the moduli space of cactus
products in one case, the Outer space of a free group in another, new spaces for the graph
products of groups and a new Outer space of a right-angled Artin group.
The categorical perspective may point the way to other possibilities. The configuration
spaces of the final chapter are currently ‘one-off’ examples which perhaps should belong
EPILOGUE 115
to a functor of similar objects. The next section sketches a possible theory.
Mapping class groups
In Chapter 3 we calculated the homology of the group FR(G1 ∗ . . . ∗Gm) which extended
results from [27] which only applies to FR(Z∗ . . .∗Z). Results about a single object have
been extended to a whole functor. In chapter 4 we studied configuration spaces of spheres
and showed that these have fundamental groups FR(Z ∗ . . . ∗ Z). This one object case
is suggestive of a theory which covers a wider range of the functor FR(−). To end this
thesis we sketch how our approach and our philosophy can be applied to mapping class
groups.
The groups FR(G1∗ . . .∗Gm) act on the free product G1∗ . . .∗Gm. Such free products
are images of the n-fold free product ∗m−1 which is a functor from (Gps)m, m copies of
the category of groups, to Gps. The plan is to replace the category of groups with a
different category n-MfldSn−1 which we describe below. The wedge product ∗m−1 will be
replaced by a different functor, the m-ary connected sum functor, denoted Wm−1. Finally
a functor of automorphisms which act on the image objects of Wm−1 is defined using
the theory of extendable automorphisms developed in Section 3.5.1. Analogues of Dehn
twists are used to construct extendable automorphisms explicitly however we make no
attempt at a full classification.
Manifolds with a chosen boundary component
In the following we will not fix a particular category of manifolds, although a default choice
could be a category of smooth manifolds of fixed dimension. However the morphisms
between manifolds will always be embeddings. The reason for not fixing a particular type
of manifold is that the theory should be well defined whatever choice is made.
Let M be an n-dimensional manifold with a chosen boundary component isomorphic
to the (n−1)-sphere
Sn−1 //M. (E.1.1)
A morphism of such manifolds will be a triangle
Sn−1 //
""E
EE
EE
EE
E M
f
N
(E.1.2)
where f is an embedding which does not necessarily need to preserve the whole boundary.
We denote this category n-MfldSn−1 . Let M be any n-manifold perhaps with boundary
and p be a point in its interior. Then by removing an open n-disc around p we get an
116 EPILOGUE
object of n-MfldSn−1 and any object can be obtained in this way.
Example E.1.1. Some objects of 2-MfldS1 :
(E.1.3)
The first is a disc with one handle attached and the only boundary component is the
chosen one. This example could have been obtained by removing a disc from a two
dimensional torus. The second is a disc with three discs removed from its interior, as
such it has four boundary components; the chosen one and the boundaries of the three
removed discs.
Example E.1.2. Examples of 3-manifolds are given by knots embedded in D3. Let
K : S1 → D3 be such a knot, then define M = D3−K(S1). Since K(S1) is closed within
D3 the manifold M only has boundary the chosen component S2. Replacing K(S1) with
a tubular open neighbourhood NK , we obtain N = D
3 − NK which is a manifold with
boundary a union of S2 and the torus S1 × S1.
There is a functor given by the fundamental group functor
pi1 : n-MfldSn−1 → Gps (E.1.4)
whenever n ≥ 3. This is because the chosen boundary sphere is simply connected and so
may be used as a ‘basepoint’. If n = 2 then boundary component is a circle and there is
no canonical choice of basepoint, if we supplied one then the target of the functor would
be the category of groups with a chosen element,
Z→ pi1(M). (E.1.5)
In the following we will restrict n to be strictly larger than two, although a theory would
still exist for surfaces.
The m-ary connected sum, Wm−1
The connected sum of objects of n-MfldSn−1 is given by identifying the two chosen
boundary components. We now describe the connected sum of m objects of n-MfldSn−1 .
EPILOGUE 117
Choose a configuration of m closed n-discs inside a large n-disc
Dn
b1
%%
bm
99D
n. (E.2.6)
These satisfy bi(D
n) ∩ bj(Dn) = ∅ if i 6= j. Now take the manifold Lnm which has the
interiors of these discs taken away so has boundary components
Sn−1
s1
''
sm
77
Lnm Sn−1.
too (E.2.7)
The map t gives the boundary of the large disc and with this t is in the category
n-MfldSn−1 . We use this manifold to define a functor
Wm−1 : (n-MfldSn−1)
m → n-MfldSn−1 (E.2.8)
as follows: take the n-manifold Lnm just defined and for each i = 1, . . . ,m glue L
n
m to Mi
along si:
Sn−1 //
si
Mi
Lnm.
(E.2.9)
This results in an n-manifold Wm−1(M1, . . . ,Mm) with chosen boundary component t.
Clearly this gluing also carries to morphisms f : Mi → Ni for i = 1, . . . ,m, so we do have
a functor Wm−1. If n ≥ 2 then there is a commuting diagram of functors
(n-MfldSn−1)
mW
m−1
//
(pi1)m
n-MfldSn−1
pi1
(Gps)m ∗
m−1
//Gps
(E.2.10)
Example E.2.3. Taking the objects from Example E.1.1 and a configuration of discs
(E.2.11)
118 EPILOGUE
the functor W 1 gives the manifold
(E.2.12)
Remark E.1. In order to take the connected sum of m objects we had to choose a config-
urations of n-discs inside a large n-disc. So in fact there is a space of possible connected
sums, although since the space of configurations is connected any two choices will give
two functors connected by a natural transformation of homeomorphisms.
The fact that there is a space of products means that we must be careful with our
definitions of monoidal categories; n-MfldSn−1 isn’t a strict monoidal category with our
definitions, but it is weakly monoidal. In particular it is not symmetric monoidal, instead
it is “En-monoidal”. When n = 2 this definition means that the category is weakly
braided monoidal.
Automorphisms extendable with respect to m-ary connected sums
Let M be an object of n-MfldSn−1 . The automorphisms of M naturally form a topo-
logical group fixing the boundary sphere Sn−1. The homotopy classes of automorphisms
pi0 Aut(M) form the mapping class group. The definition of an extendable automorphism
is contained in Section 3.5.1.
Definition E.2. Let MFR(M1, . . . ,Mm) be the topological group of extendable auto-
morphisms of Wm−1(M1, . . . ,Mm).
Remark E.3. Recall that the extendability of an automorphism α of Wm−1(M1, . . . ,Mm)
says that for every m-tuple of morphisms fi : Mi → Ni the diagram
Wm−1(M1, . . . ,Mm)
Wm−1(f1,...,fm)
α //Wm−1(M1, . . . ,Mm)
Wm−1(f1,...,fm)
Wm−1(N1, . . . , Nm) β
//Wm−1(N1, . . . , Nm)
(E.3.13)
may be filled in to a commuting square by some automorphism β of Wm−1(N1, . . . , Nm).
Such extendable automorphisms in MFR(M1, . . . ,Mm) may be constructed by choos-
ing a method to build automorphisms by using only a minimal amount of information
attached to Wm−1(M1, . . . ,Mm). The information will be mapped by any m-tuple of
EPILOGUE 119
morphisms fi : Mi → Ni to Wm−1(N1, . . . , Nm) where it may be used to construct an
automorphism in MFR(N1, . . . , Nm) which extends the original one.
Twists through tubes
In the following a tube Tn will be a copy of D
n−1×S1 whose points may be parametrised
by
(n, r, t) , (E.4.14)
where n ∈ Sn−2, r ∈ [0, 1] and t ∈ S1 ∼= [0, 2pi] /(0 ∼ 2pi). The map
Sn−2 × [0, 1]→ Dn−1 ∼= {x ∈ Rn−1 | |x| ≤ 1} (E.4.15)
is defined by (n, r) 7→ r.n. The core Cn of Tn is the set of points with r ≤ 12 . We now
define a twisting map φ : Tn → Tn which fixes the core Cn and the boundary of Tn, but
‘twists’ the remaining part of the tube as follows
ϕ(n, r, t) =
(n, r, t) if |r| ≤ 12 , that is a point in the core,(n, r, t+ 2pi(2r − 1)) otherwise. (E.4.16)
This is a homeomorphism, the inverse can be given just by changing a sign.
Example E.4.4. When n = 2 the tube Tn is an annulus and the twisting map ϕ is
represented below. The vertical lines in the left diagram are taken to the spiralling lines
in the right.
(E.4.17)
Note that if one restricts attention to the core which is bounded by the dashed lines, one
sees that the regions coloured grey are identical, this is because ϕ is the identity on the
core.
Definition E.4. A tube with support on M2, . . . ,Mm is an embedding E of Tn into
Wm−1(Dn,M2, . . . ,Mm) such that the copy of Dn in the first position is contained within
the core E(Cn).
120 EPILOGUE
The automorphism αE ∈ Aut(Wm−1(M1, . . . ,Mm)) associated to E is defined using
the tube twist ϕ as follows
αE(y) =
y if y ∈M1,
y if y /∈ E(Tn),
E(ϕ(y)) if y = E(x).
(E.4.18)
Proposition E.5. Let E(Tn) be a tube with support on M2, . . . ,Mm. Then αE is extend-
able with respect to Wm−1.
Proof. Suppose (fi) is an m-tuple of morphisms fi : Mi → Ni in n-MfldSn−1 . Then
Wm−1(IDn , f2, . . . , fm) is an embedding ofWm−1(Dn,M2, . . . ,Mm) intoWm−1(Dn, N2, . . . , Nm).
ThereforeWm−1(IDn , f2, . . . , fm)◦E is an embedding of a tube intoWm−1(Dn, N2, . . . , Nm).
Since Dn is fixed by Wm−1(IDn , f2, . . . , fm) it is also contained in the image of the tube
embedding so Wm−1(IDn , f2, . . . , fm) ◦ E gives a tube with support N2, . . . , Nm, we will
denote the embedding f(E). It is an easy check that the square
Wm−1(M1, . . . ,Mm)
αE //
Wm−1(f1,...,fm)
Wm−1(M1, . . . ,Mm)
Wm−1(f1,...,fm)
Wm−1(N1, . . . , Nm)
αf(E) //Wm−1(N1, . . . , Nm)
(E.4.19)
commutes, so αE is extendable.
The automorphisms αE are analogues of partial conjugations: for n > 2 the maps
pi1(αE) are partial conjugations of the groups pi1(W
m−1(M1, . . . ,Mm)), which are free
products of the groups pi1(Mi;S
n−1).
It is worth noting that the functor pi1 takes automorphisms to automorphisms, however
there is no guarantee that the image of an extendable automorphism is itself extendable.
So we can not conclude that every extendable automorphism induces a Whitehead auto-
morphism.
Remark E.6. Suppose we were to restrict the morphisms in the category n-MfldSn−1
to be homeomorphisms. Then every automorphism of Wm−1(M1, . . . ,Mm) would be ex-
tendable because the morphisms they would extend along would be automorphisms and
hence the extension is given by conjugation. This should serve as a warning; if Mi is a
compact manifold where the whole boundary is Sn−1 then the embeddings out of Mi are
isomorphisms, hence Mi supplies no interesting morphisms to extend along and so the
extendability problem is different from that of the theory for groups.
A further remark: we were able to show that the extendable morphisms with respect
to the free product functor were precisely the Whitehead automorphisms. The proof
relied on a lemma of Schupp 3.5.3 from [43] which says that every group is embeddable
EPILOGUE 121
into a malnormal subgroup of a complete group. We do not yet know an analogue of this
Lemma in the category n-MfldSn−1 .
The classifying space of MFR(M1, . . . ,Mm)
Just as the group FR(G1 ∗ . . . ∗ Gm) has a classifying space given by the moduli space
of cactus products, a space of gluings of manifolds should provide a classifying space for
MFR(M1 ∗ . . .∗Mm). An example is given in the first half of the previous chapter, where
the manifolds considered are closed n-discs with an unknotted (n−2)-sphere removed,
see Example E.1.2. We studied spaces of configurations of these (n−2)-spheres in Rn
(note that we have changed our dimension conventions slightly) and these configuration
spaces should be related to the classifying space of MFR(Dn − Sn−2, . . . , Dn − Sn−2).
This inspires the following conjecture.
Conjecture. The topological group MFR(M1, . . . ,Mm) has a classifying space with the
homotopy type of a finite dimensional CW complex. Furthermore the homology
H∗(BMFR(M1, . . . ,Mm),Z) (E.5.20)
splits into a direct sum over forest posets ⊕
forest poset F
. (E.5.21)
122 4. CONFIGURATION SPACES
References
[1] H. Abels and S. Holz. Higher generation by subgroups. Journal of Algebra,
160(2):310–341, 1993.
[2] J.C. Baez, D.K. Wise, and A.S. Crans. Exotic statistics for strings in 4d BF theory.
Advances in Theoretical and Mathematical Physics, 11(5):707, 2007.
[3] A. Baudisch. Subgroups of semifree groups. Acta Mathematica Hungarica, 38(1):19–
28, 1981.
[4] E. Berkove and J. Meier. The cohomology groups of the outer Whitehead automor-
phism group of a free product. Forum Mathematicum, 22(2):379–395, 2010.
[5] N. Brady, J. McCammond, J. Meier, and Miller A. The pure symmetric automor-
phisms of a free group form a duality group. Journal of Algebra, 246(2):881–896,
2001.
[6] T. Brendle and A. Hatcher. Configuration spaces of rings and wickets. Commentarii
Mathematici Helvetici, In Press, 2012.
[7] K.S. Brown. Cohomology of Groups. Graduate Texts in Mathematics. Springer-
Verlag, 1982.
[8] K.S. Brown. The coset poset and probabilistic zeta function of a finite group. Journal
of Algebra, 225(2):989–1012, 2000.
[9] A. Brownstein and R. Lee. Cohomology of the group of motions of n strings in 3-
space. In Mapping class groups and moduli spaces of Riemann surfaces (Go¨ttingen,
1991/Seattle, WA, 1991), volume 150, pages 51–61. American Mathematical Society,
1993.
[10] R. Charney. An introduction to right-angled Artin groups. Geometriae Dedicata,
125(1):141–158, March 2007.
[11] R. Charney, K. Ruane, N. Stambaugh, and A. Vijayan. The automorphism group of
a graph product with no SIL. Illinois J. Math., 54(1):249–262, 2010.
123
124 REFERENCES
[12] R. Charney and K. Vogtmann. Finiteness properties of automorphism groups of
right-angled Artin groups. Bull. Lond. Math. Soc., 41(1):94–102, 2009.
[13] Y. Chen, H. Glover, and C. Jensen. Proper actions of automorphism groups of
free products of finite groups. International Journal of Algebra and Computation,
15(2):255–272, 2005.
[14] D.J. Collins. Cohomological dimension and symmetric automorphisms of a free group.
Commentarii Mathematici Helvetici, 64(1):44–61, 1989.
[15] D.J. Collins and H. Zieschang. Rescuing the whitehead method for free products, i:
Peak reduction. Mathematische Zeitschrift, 185:487–504, 1984.
[16] D.J. Collins and H. Zieschang. Rescuing the whitehead method for free products, ii:
The algorithm. Mathematische Zeitschrift, 186:335–361, 1984.
[17] M. Culler and K. Vogtmann. Moduli of graphs and automorphisms of free groups.
Inventiones mathematicae, 84(1):91–119, 1986.
[18] D.M. Dahm. A generalization of braid theory. Princeton Univ. Ph. D. thesis, 1962.
[19] M.B. Day. Peak reduction and finite presentations for automorphism groups of right-
angled Artin groups. Geom. Topol., 13(2):817–855, 2009.
[20] V. Dotsenko and A. Khoroshkin. Gro¨bner bases for operads. Duke Math. J.,
153(2):363–396, 2010.
[21] D.I. Fouxe-Rabinovitch. U¨ber die Automorphismengruppen der freien Produkte. I.
Mat. Sbornik, 8(1):264–276, 1940.
[22] E.R. Green. Graph products of groups. Univ. of Leeds Ph. D. thesis, 1990.
[23] V. Guirardel and G. Levitt. The outer space of a free product. Proceedings of the
London Mathematical Society, 94(3):695, 2007.
[24] M. Gutierrez, A. Pigott, and K. Ruane. On the automorphisms of a graph product
of abelian groups. To appear in Groups Geom. Dyn., 2011.
[25] A. Hatcher. Algebraic topology. Cambridge University Press, 2002.
[26] H. Helmholtz. U¨ber Integrale der hydrodynamischen Gleichungen, welche den
Wirbelbewegungen entsprechen. Journal fu¨r die reine und angewandte Mathematik,
1858.
[27] C. Jensen, J. McCammond, and J. Meier. The integral cohomology of the group of
loops. Geometry & Topology, 10:759–784, 2006.
REFERENCES 125
[28] C. Jensen, J. McCammond, and J. Meier. The Euler characteristic of the Whitehead
automorphism group of a free product. Transactions of the American Mathematical
Society, 359(6):2577, 2007.
[29] C. Jensen and N. Wahl. Automorphisms of free groups with boundaries. Algebr.
Geom. Topol, 4:543–569, 2004.
[30] Lord Kelvin. On vortex atoms. Proceedings of the Royal Society of Edinburgh, VI:94–
105, 1867.
[31] J. Leach, M.R. Dennis, J. Courtial, and M.J. Padgett. Laser beams: knotted threads
of darkness. Nature, 432(7014):165, 2004.
[32] I.J. Leary. On the integral cohomology of wreath products. Journal of Algebra,
198(1):184–239, 1997.
[33] D. Margalit and J. McCammond. Geometric presentations for the pure braid group.
J. Knot Theory Ramifications, 18(1):1–20, 2009.
[34] M. Markl, S. Shnider, and J. Stasheff. Operads in algebra, topology and physics, vol-
ume 96 of Mathematical Surveys and Monographs. American Mathematical Society,
Providence, RI, 2002.
[35] J. McCammond and J. Meier. The hypertree poset and the l 2-Betti numbers of the
motion group of the trivial link. Mathematische Annalen, 328(4):633–652, 2004.
[36] J. McCool. On basis-conjugating automorphisms of free groups. Canadian J. Math,
38:1525–1529, 1986.
[37] D. McCullough and A. Miller. Symmetric automorphisms of free products. American
Mathematical Society, 1996.
[38] E. Meucci. Relative outer automorphisms of free groups. Arxiv preprint
arXiv:1010.4753, 2010.
[39] G. Mislin and I. Chatterji. Guido’s book of conjectures: a gift to Guido Mislin on
the occasion of his retirement from ETHZ, June 2006. Enseignement mathe´matique,
2008.
[40] K. Moffatt. Vortex dynamics: the legacy of Helmholtz and Kelvin. In IUTAM
Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, pages 1–10.
Springer, 2008.
[41] A. Pettet. Finiteness properties for a subgroup of the pure symmetric automorphism
group. C. R. Math. Acad. Sci. Paris, 348(3-4):127–130, 2010.
126 REFERENCES
[42] R.L. Rubinsztein. On the group of motions of oriented, unlinked and unknotted
circles in R3, I. Preprint, 2002.
[43] P.E. Schupp. A characterization of inner automorphisms. Proceedings of the American
Mathematical Society, 101(2):226–228, 1987.
[44] R.P. Stanley. Enumerative combinatorics: Volume 2. Cambridge University Press,
2001.