The sutured Floer homology polytope

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 $P\left(M,\gamma \right)$ in $H^2\left(M,\partial M;ℝ\right)$ that is spanned by the ${Spin}^c$–structures which support nonzero Floer homology groups. If $\left(M,\gamma \right)⇝\left(M^{\prime },{\gamma }^{\prime }\right)$ is a taut surface decomposition, then an affine map projects $P\left(M^{\prime },{\gamma }^{\prime }\right)$ onto a face of $P\left(M,\gamma \right)$; moreover, if $H_2\left(M\right)=0$, then every face of $P\left(M,\gamma \right)$ can be obtained in this way for some surface decomposition. We show that if $\left(M,\gamma \right)$ is reduced, horizontally prime and $H_2\left(M\right)=0$, then $P\left(M,\gamma \right)$ is maximal dimensional in $H^2\left(M,\partial M;ℝ\right)$. This implies that if $rk\left(SFH\left(M,\gamma \right)\right)<2^{k+1}$, then $\left(M,\gamma \right)$ 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.