The sutured Floer homology polytope

TL;DR

This paper extends sutured Floer homology theory by defining a polytope that encodes non-zero Floer homology structures, linking surface decompositions to polytope faces and relating Floer homology rank to manifold complexity.

Contribution

The paper introduces a polytope in cohomology supporting non-zero Floer homology, connecting surface decompositions to polytope faces and analyzing the complexity of sutured manifolds.

Findings

The polytope P(M,g) is spanned by Spin^c-structures with non-zero Floer homology.

Surface decompositions project the polytope onto faces, characterizing manifold structures.

The rank of Floer homology bounds the topological complexity of the manifold.

Abstract

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.