Rabinowitz-Floer homology on Brieskorn manifolds

TL;DR

This thesis develops Rabinowitz-Floer homology for fillable contact structures on odd-dimensional manifolds, addressing technical challenges, proving invariance properties, and computing RFH for Brieskorn manifolds to understand contact structure classifications.

Contribution

It advances the definition and computation of Rabinowitz-Floer homology on Brieskorn manifolds, establishing invariance under handle attachment and providing new insights into fillable contact structures.

Findings

RFH is essentially invariant under subcritical handle attachment

RFH can be computed explicitly for certain Brieskorn manifolds

Results imply either infinitely many fillable contact structures or infinite-dimensional RFH in each degree

Abstract

This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports fillable contact structures, that there exist either infinitely many different fillable contact structures, or one contact structure with infinitely many different fillings or for every fillable contact structure holds that RFH is infinite dimensional in every degree.