A refinement of sutured Floer homology

Abstract

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,\tau )$ by Juhasz. An algebra $A_{\tau }$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\mathbb{Z})$ is defined. For a fixed Spin^c structure $s$ over the manifold $X^{\prime }$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,\tau ,s)$ is then defined as a chain complex with coefficients in $A_{\tau }$ and filtered by $\SpinC(X,τ)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,\tau )$ and the Spin^c class $s\in\SpinC(X')$. The construction generalizes the construction of Juhasz. It plays the role of $C{F}^{-}(X,s)$ when $X$ is a closed three-manifold, and the role of $CF{K}^{-}(Y,K;s)$ when the sutured manifold is obtained from a knot $K$ inside a three-manifold $Y$. Our invariants generalize both the knot invariants of Ozsvath-Szabo and Rasmussen and the link invariants of Ozsvath and Szabo. We study some of the basic properties of the corresponding Ozsvath-Szabo complex, including the exact triangles, and some form of stabilization.