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