Symplectic homology and the Eilenberg-Steenrod axioms

Abstract

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.

v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topology

References in corpus (1)

• Symplectic and contact differential graded algebras

Cited by in corpus (28)

• Symplectic capacities from positive S^1-equivariant symplectic homology
• Floer theory and reduced cohomology on open manifolds
• From symplectic cohomology to Lagrangian enumerative geometry
• Rabinowitz Floer homology and mirror symmetry
• Mayer-Vietoris property for relative symplectic cohomology
• Vanishing of Symplectic Homology and Obstruction to Flexible Fillability
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds
• $({\mathbb{RP}}^{2n-1},{\xi }_{std})$ is not exactly fillable for $n\ne 2^k$
• Two closed orbits for non-degenerate Reeb flows
• Categorical mirror symmetry on cohomology for a complex genus 2 curve
• Simplicial descent for Chekanov-Eliashberg dg-algebras
• On filings of $\partial (V\times \mathbb{D})$
• On fillings of contact links of quotient singularities
• The persistence of a relative Rabinowitz-Floer complex
• Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence
• On the cohomology ring of symplectic fillings
• Reduced symplectic homology and string topology
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Contact open books with flexible pages
• The spectral diameter of a Liouville domain
• Computing the Rabinowitz Floer homology of tentacular hyperboloids
• Legendrian non-isotopic unit conormal bundles in high dimensions
• On weakly exact Lagrangians in Liouville bi-fillings
• V-shaped action functional with delay
• Calabi-Yau structure on the Chekanov-Eliashberg algebra of a Legendrian sphere
• Invariant Sets and Hyperbolic Closed Reeb Orbits