A refinement of sutured Floer homology

We introduce a refinement of the Ozsváth–Szabó complex associated by Juhász [Ju1] to a balanced sutured manifold $(X, \tau)$. An algebra $\mathbb{A}_{\tau}$ is associated to the boundary of a sutured manifold. For a fixed class $\mathfrak{s}$ of a $\mathrm{Spin}^c$ structure over the manifold $\overline{X}$, which is obtained from $X$ by filling out the sutures, the Ozsváth–Szabó chain complex $\mathrm{CF}(X, \tau, \mathfrak{s})$ is then defined as a chain complex with coefficients in $\mathbb{A}_{\tau}$ and filtered by the relative $\mathrm{Spin}^c$ classes in $\mathrm{Spin}^c (X, \tau)$. The filtered chain homotopy type of this chain complex is an invariant of $(X, \tau)$ and the $\mathrm{Spin}^c$ class $\mathfrak{s} \in \mathrm{Spin}^c (\overline{X})$. The construction generalizes the construction of Juhász. It plays the role of $\mathrm{CF}^{-} (X, \mathfrak{s})$ when $X$ is a closed three-manifold, and the role of\[\mathrm{CFK}^{-} (Y, K; \mathfrak{s}) = \bigoplus_{\underline{\mathfrak{s}} \in \mathfrak{s}} \mathrm{CFK}^{-} (Y, K, \underline{\mathfrak{s}}) ,\]when the sutured manifold is obtained from a knot $K$ inside a three-manifold $Y$. Our invariants thus generalize both the knot invariants of Ozsváth–Szabó and Rasmussen and the link invariants of Ozsváth and Szabó. We study some of the basic properties of the Ozsváth–Szabó complex corresponding to a balanced sutured manifold, including the behaviour under boundary connected sum, some form of stabilization for the complex, and an exact triangle generalizing the surgery exact triangles for knot Floer complexes.

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Published 5 October 2015