Duality in filtered Floer-Novikov complexes

TL;DR

This paper establishes a duality pairing in filtered Floer-Novikov complexes that is nondegenerate on homology, extending previous results and enabling new applications in symplectic topology.

Contribution

It proves a nondegenerate duality pairing for filtered Floer complexes and extends the validity of a Calabi quasimorphism construction without rationality assumptions.

Findings

Proves nondegeneracy of a bilinear pairing in Floer complexes

Extends duality results to non-rational symplectic manifolds

Provides a new formula for boundary depth invariance

Abstract

We prove that a certain bilinear pairing (analagous to the Poincare-Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh-Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain rational symplectic manifolds, and the result that we prove here implies that their construction remains valid when the rationality hypothesis is dropped. Apart from this, we also use the nondegeneracy of the pairing to establish a new formula for what we have previously called the boundary depth of a Floer chain complex; this formula shows that the boundary depth is unchanged under passing to the opposite complex.