The sutured Floer polytope and taut depth-one foliations

Abstract

For closed 3–manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3–manifold $Y$, there is a bijection between vertices of the ${HF}^{+}\left(Y\right)$ polytope carrying the group $\mathbb{Z}$ and the faces of the Thurston norm unit ball that correspond to fibrations of $Y$ over the unit circle. Moreover, the Thurston norm unit ball of $Y$ is dual to the polytope of $\widehat{\underset{¯}{HF}}\left(Y\right)$.

We prove a similar bijection and duality result for a class of 3–manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two possibly disconnected surfaces with boundary $R_{+}$ and $R_{-}$. We show that there is a bijection between vertices of the sutured Floer polytope carrying the group $\mathbb{Z}$ and equivalence classes of taut depth-one foliations that form the foliation cones of Cantwell and Conlon. Moreover, we show that a function defined by Juhász, which we call the geometric sutured function, is analogous to the Thurston norm in this context. In some cases, this function is an asymmetric norm and our duality result is that appropriate faces of this norm’s unit ball subtend the foliation cones.

An important step in our work is the following fact: a sutured manifold admits a fibration or a taut depth-one foliation whose sole compact leaves are exactly the connected components of $R_{+}$ and $R_{-}$, if and only if, there is a surface decomposition of the sutured manifold resulting in a product manifold.