Rabinowitz–Floer Homology on Brieskorn Spheres: Fig. 1.

Abstract

SelbstständigkeitserklärungHiermit erkläre ich, dass ich die vorliegende Dissertation selbsständig und nur unter Verwendung der angegebenen Literatur und Hilfsmittel angefertig habe.i ii Acknowledgements I wish to thank my supervisor Klaus Mohnke, who introduced me to the field of symplectic and contact geometry. The discussions with him were at the same time inspiring and encouraging. Without him insisting on clear explanations, some parts of this thesis would never have been written.

Maximum principles in symplectic homology

“…for large r, where h t : Σ → R is a function which is invariant under the Reeb flow of α. A similar construction was also given independently by the second author in [26], and for time-independent h a different proof was given by Fauck in his thesis [9].…”

Bounds for tentacular Hamiltonians

“…In [2] and [15] the Rabinowitz Floer chain complex has been related to the Morse (co)chain complex, and this resulted in the computation of the Rabinowitz Floer homology of some hypersurfaces in twisted cotangent bundles. RFH of Brieskorn manifolds has been studied in [11], and used to study fillable contact structures on closed manifolds. Moreover, RFH has been linked to the existence of leaf-wise intersection points [3], via a particular perturbation of the Rabinowitz action functional, and to questions of orderability and non-squeezing in symplectic geometry [4].…”

Symplectic homology of some Brieskorn manifolds

“…Later on, van Koert [38] calculated the cylindrical contact homology for all Brieskorn manifolds for which it is (conjecturally) well-defined, using Morse-Bott methods from [3]. In another event, Fauck [14] has reproven Ustilovsky's theorem using Rabinowitz Floer homology, which has the advantage that its analytic foundations are well-established.…”

“…For n odd and k even, the contact structures are distinguished in [37] and [14]. Note that their results can also be proven using symplectic homology, with computations almost identical to [14]. This leaves the case n odd and k odd, which is treated here.…”