The decategorification of sutured Floer homology

摘要

We define a torsion invariant T for every balanced sutured manifold (M, γ), 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, γ) is complementary to a Seifert surface of an alternating knot, then SFH(M, γ) is either 0 or Z in every Spine structure. The torsion invariant T can also be used to show that a sutured manifold is not disc decomposable, and to distinguish between Seifert surfaces. The support of SFH gives rise to a norm z on H_2(M, OM; R). The invariant T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x~8. For closed 3-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.