  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.
