Symplectic homology and the Eilenberg–Steenrod axioms

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg–Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer–Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology.

In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.

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Keywords

Floer homology, symplectic homology, contact homology, Rabinowitz–Floer homology, Eilenberg–Steenrod axioms for a homology theory, Liouville cobordisms

Mathematical Subject Classification 2010

Primary: 53D40, 55N40, 57R17

Secondary: 57R90

References

Publication

Received: 28 June 2016

Revised: 9 February 2018

Accepted: 21 February 2018

Published: 26 April 2018

Authors

Kai Cieliebak
Universität Augsburg Augsburg Germany

Alexandru Oancea
Sorbonne Université Université Paris Diderot CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche IMJ-PRG Paris France

Kai Cieliebak


Alexandru Oancea


Peter Albers
Mathematisches Institut Universität Heidelberg Heidelberg Germany

Volume 18, issue 4 (2018)

Kai Cieliebak and Alexandru Oancea

Appendix: Kai Cieliebak, Alexandru Oancea and Peter Albers

Abstract