A refinement of sutured Floer homology

We introduce a refinement of the Ozsvath-Szabo complex associated to a balanced sutured manifold $(X,τ)$ by Juhasz. An algebra $A_τ$ is associated to the boundary of a sutured manifold and a filtration of its generators by $H^2(X,\partial X;\Z)$ is defined. For a fixed Spin^c structure $s$ over the manifold $X'$, which is obtained from $X$ by filling out the sutures, the Ozsvath-Szabo chain complex $CF(X,τ,s)$ is then defined as a chain complex with coefficients in $A_τ$ and filtered by $\SpinC(X,τ)$. The filtered chain homotopy type of this chain complex is an invariant of $(X,τ)$ and the Spin^c class $s\in\SpinC(X')$. The construction generalizes the construction of Juhasz. It plays the role of $CF^-(X,s)$ when $X$ is a closed three-manifold, and the role of $CFK^-(Y,K;s)$ 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.

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