Projective twists and the Hopf correspondence

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Torricelli, Brunella Charlotte 

This dissertation is the fruit of a research project on a class of symplectic automorphisms called projective twists.

In the first part of the thesis (Chapters 3,4) we use PicardLefschetz theory to introduce a new local model for the planar projective twists τAP2Sympct(TAP2), A∈{R,C}. In each case, we construct an exact Lefschetz fibration π:TAP2→C with three singular fibres, and define a compactly supported symplectomorphism φSympct(TAP2) on the total space.

Given two disjoint Lefschetz thimbles Δα,ΔβTAP2, we compute the Floer cohomology groups HF(φk(Δα),Δβ;Z/2Z) and verify (partially for CP2) that φ is indeed isotopic to (a power of) the standard local projective twist.

The constructions we present are governed by generalised lantern relations, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an S1-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of (TCP2,dλTCP2).

In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the Hopf correspondence, a Lagrangian correspondence (in the sense of WehrheimWoodward) aimed at assigning Lagrangian spheres L1,…,Lm of a Liouville manifold (Y,Ω) to given Lagrangian (real, complex) projective spaces K1,…,Km of a Liouville manifold (W,ω). When this correspondence can be established, it intertwines the (real, complex) projective twists τKiπ0(Sympct(W)) (and the induced autoequivalences of the compact Fukaya category Fuk(W)) with the Dehn twists τLiπ0(Sympct(Y)) (and the corresponding autoequivalences of Fuk(Y)), for i=1,…m. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds.

The same techniques are also used to show that in infinitely many dimensions n, the Hamiltonian class of the local projective twist in Sympct(TCPn) does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces KCPn, that do not admit Lagrangian embeddings into (TCPn,dλTCPn), for n=4,7.

Smith, Ivan
symplectic topology, Dehn twist, Floer cohomology
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
EPSRC (1804059)
EPSRC studentship