On the equivalence of contact invariants in sutured Floer homology theories

TL;DR

This paper establishes an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology, linking their contact invariants and demonstrating functoriality of Legendrian invariants in knot Floer homology.

Contribution

It proves the equivalence of contact invariants in two Floer homology theories and shows their functorial behavior under Lagrangian concordance, providing new computational tools.

Findings

Isomorphism between sutured monopole and Heegaard Floer homology.

Identification of contact invariants across theories.

Functoriality of Legendrian invariants under Lagrangian concordance.

Abstract

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Mati\'c's contact class is a well-defined invariant of contact manifolds.