Symplectic homology and the Eilenberg-Steenrod axioms

TL;DR

This paper develops a symplectic homology theory for pairs of Liouville cobordisms, satisfying axioms similar to Eilenberg-Steenrod, and introduces new exact sequences, invariance results, and algebraic structures.

Contribution

It defines symplectic homology for Liouville cobordisms satisfying Eilenberg-Steenrod-like axioms and establishes new exact sequences and invariance properties.

Findings

Established a Mayer-Vietoris sequence for symplectic homology

Proved invariance of Rabinowitz Floer homology under subcritical handle attachment

Introduced a new product unifying pair-of-pants and coproduct structures

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.