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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.
Keywords: Floer homology, symplectic homology, contact homology, Rabinowitz–Floer homology, Eilenberg–Steenrod axioms for a homology theory, Liouville cobordisms
Cieliebak, Kai 1 ; Oancea, Alexandru 2
@article{10_2140_agt_2018_18_1953,
author = {Cieliebak, Kai and Oancea, Alexandru},
title = {Symplectic homology and the {Eilenberg{\textendash}Steenrod} axioms},
journal = {Algebraic and Geometric Topology},
pages = {1953--2130},
publisher = {mathdoc},
volume = {18},
number = {4},
year = {2018},
doi = {10.2140/agt.2018.18.1953},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1953/}
}
TY - JOUR AU - Cieliebak, Kai AU - Oancea, Alexandru TI - Symplectic homology and the Eilenberg–Steenrod axioms JO - Algebraic and Geometric Topology PY - 2018 SP - 1953 EP - 2130 VL - 18 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1953/ DO - 10.2140/agt.2018.18.1953 ID - 10_2140_agt_2018_18_1953 ER -
%0 Journal Article %A Cieliebak, Kai %A Oancea, Alexandru %T Symplectic homology and the Eilenberg–Steenrod axioms %J Algebraic and Geometric Topology %D 2018 %P 1953-2130 %V 18 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.1953/ %R 10.2140/agt.2018.18.1953 %F 10_2140_agt_2018_18_1953
Cieliebak, Kai; Oancea, Alexandru. Symplectic homology and the Eilenberg–Steenrod axioms. Algebraic and Geometric Topology, Tome 18 (2018) no. 4, pp. 1953-2130. doi: 10.2140/agt.2018.18.1953
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