Université de Nantes - Université Paris-Saclay

NANTES-ORSAY SEMINAR ON
SYMPLECTIC AND CONTACT GEOMETRY

Friday, 9 December 2022

Vera Vertesi (Vienna)

A contact invariant in bordered Floer homology

Abstract: Many of the recent advances in classifying tight contact structures were made possible by the advent of Heegaard Floer homology in the early 2000s and the subsequent development of Floer theoretic contact invariants. Using open books, Ozsváth and Szabó defined an invariant of closed contact three-manifolds. This "contact class" was used to show that knot Floer homology detects both genus (Ozsváth-Szabó) and fiberedness (Ghiggini, Ni). It also gives information about overtwistedness: the contact class vanishes for overtwisted contact structures, and does not vanish for Stein fillable ones (Ozsváth-Szabó). The contact class was also used to distinguish notions of fillability: Ghiggini used it to construct examples of strongly symplectically fillable contact three-manifolds which do not have Stein fillings. In this talk I define a relative version of the contact class for contact manifolds with "decorated" boundary, and explain how this can be used to keep track of the contact invariant while building it up from elementary pieces. This is a joint work with Akram Alishahi, Viktória Földvári, Kristen Hendricks, Joan Licata and Ina Petkova.

Paolo Ghiggini (Grenoble and CNRS)

Bordered Floer modules from compact objects and genus two mutation

Abstract: Bordered Floer homology is an invariant of three-manifolds with boundary introduced by Lipshitz, Ozsváth and Thurston. To every closed, compact and connected surface F equipped with a preferred handle decomposition it associates a differential graded algebra A(F) and to every compact, connected and oriented three-manifold M with boundary F it associates an A-infinity module CFA(M) over A(F). Moreover, if a closed three-manifold Y is decomposed in two pieces by a surface, then the Heegaard Floer homology of Y can be recovered from the morphisms between the CFA modules of the two pieces. Auroux proved that the algebra A(F) can be interpreted as the morphism spaces of a set of generators of the partially wrapped Fukaya category of a symmetric product of F minus a point. I will show how to associate an object X(M) in the triangulated envelope of the compact Fukaya category of a symmetric product of F minus a point to a three-manifold M with boundary F such that the module CFA(M) is quasi-isomorphic to the morphisms in the partially wrapped Fukaya category between X(M) and Auroux's generators. As an application, I will show that a surgery operation on three-manifold called genus two mutation does not change the total rank of Heegaard Floer homology.

Discussions