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Symplectic homology and the Eilenberg–Steenrod axioms

Kai Cieliebak and Alexandru Oancea

Appendix: Kai Cieliebak, Alexandru Oancea and Peter Albers

Algebraic & Geometric Topology 18 (2018) 1953–2130

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.

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
Received: 28 June 2016
Revised: 9 February 2018
Accepted: 21 February 2018
Published: 26 April 2018
Kai Cieliebak
Universität Augsburg
Alexandru Oancea
Sorbonne Université
Université Paris Diderot
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Kai Cieliebak
Alexandru Oancea
Peter Albers
Mathematisches Institut
Universität Heidelberg