## Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy

##### Authors

Nordström, Johannes

##### Date

2008-10-14##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Nordström, J. (2008). Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy (Doctoral thesis). https://doi.org/10.17863/CAM.16204

##### Abstract

In Berger's classification of Riemannian holonomy groups
there are several infinite families and two exceptional cases:
the groups Spin(7) and G_2.
This thesis is mainly concerned with 7-dimensional manifolds
with holonomy G_2.
A metric with holonomy contained in G_2 can be defined in terms of
a torsion-free G_2-structure, and a G_2-manifold is a 7-dimensional manifold
equipped with such a structure.
There are two known constructions of compact manifolds with holonomy
exactly G_2. Joyce found examples by resolving singularities of
quotients of flat tori.
Later Kovalev found different examples by gluing pairs of exponentially
asymptotically cylindrical (EAC) G_2-manifolds (not necessarily with holonomy
exactly G_2) whose cylinders match. The result of this gluing construction
can be regarded as a generalised connected sum of the EAC components, and has
a long approximately cylindrical neck region.
We consider the deformation theory of EAC G_2-manifolds and show, generalising from
the compact case, that there is a smooth moduli space of torsion-free EAC
G_2-structures.
As an application we study the deformations of the gluing construction for
compact G_2-manifolds, and find that the glued torsion-free G_2-structures form an open
subset of the moduli space on the compact connected sum. For a fixed pair of
matching EAC G_2-manifolds the gluing construction provides a path of torsion-free
G_2-structures on the connected sum with increasing neck length.
Intuitively this defines a boundary point for the moduli space on the connected
sum, representing a way to `pull apart' the compact G_2-manifold into a pair of EAC
components. We use the deformation theory to make this more precise.
We then consider the problem whether compact G_2-manifolds constructed by Joyce's
method can be deformed to the result of a gluing construction.
By proving a result for resolving singularities of EAC G_2-manifolds we show
that some of Joyce's examples can be pulled apart in the above sense.
Some of the EAC G_2-manifolds that arise this way satisfy a necessary and
sufficient topological condition for having holonomy exactly G_2.
We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8
manifolds with holonomy contained in Spin(7). On such manifolds there is
a smooth moduli space of torsion-free EAC Spin(7)-structures.
Generalising a result of Wang for compact manifolds we show that for
EAC G_2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.

##### Keywords

differential geometry, special holonomy

##### Sponsorship

This work was supported by the EPSRC and the Gates Cambridge Trust.