The decategorification of sutured Floer homology

TL;DR

This paper introduces a torsion invariant T for balanced sutured manifolds, showing it aligns with sutured Floer homology's Euler characteristic, and uses it to derive new insights into the topology of these manifolds and their Seifert surfaces.

Contribution

It defines a new torsion invariant T that simplifies computations and provides new tools for analyzing sutured Floer homology and related topological properties.

Findings

T equals the Euler characteristic of SFH and is computable via Fox calculus.

T helps determine when SFH is trivial or Z in each spin^c structure.

T provides bounds on the sutured Thurston norm and distinguishes between Seifert surfaces.

Abstract

We define a torsion invariant T for every balanced sutured manifold (M,g), and show that it agrees with the Euler characteristic of sutured Floer homology SFH. The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M,g) is complementary to a Seifert surface of an alternating knot, then SFH(M,g) is either 0 or Z in every spin^c structure. T can also be used to show that a sutured manifold is not disk decomposable, and to distinguish between Seifert surfaces. The support of SFH gives rise to a norm z on H_2(M, \partial M; R). Then T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x^s. For closed three-manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z < x^s can happen in general. Finally, we compute T for several wide classes of sutured manifolds.