Sutured Floer homology, fibrations, and taut depth one foliations (Q2790733)

Heegaard Floer homology was defined by Ozsváth and Szabó and is a package of invariants for closed 3-manifolds. Sutured Floer homology (SFH) defined by Juhász is a generalisation of the hat versions of Heegaard Floer homology to 3-manifolds with boundary decorated with a dividing set, called sutured manifolds. In the paper under review the authors prove that SFH detects whether or not an irreducible \(3\)-manifold \(M\) with non-empty boundary is fibred, and determines all fibred classes in the first cohomology group of \(M\). In addition, given a connected, irreducible balanced sutured manifold \((M,\gamma)\), and \(\alpha \in H^1(M)\), the authors prove that \(SFH_{\alpha}(M,\gamma)\simeq \mathbb Z\) if and only if there exists an indecomposable taut depth one foliation \(\mathcal F\) with \(\lambda (\mathcal F) = \alpha\). Here \(\lambda (\mathcal F)\) is a cohomology class in \(H^1(M)\) that can be naturally associated to an indecomposable taut depth one foliation \(\mathcal F\) on \((M,\gamma)\). Note that the last result was also proven by Altman some time ago for strongly balanced sutured manifolds \(M\) with \(H_2(M) = 0\).