Sutured Floer homology, fibrations, and taut depth one foliations

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology (SFH) can be used to determine all fibred classes in H1(M). Furthermore, we show that the SFH of a balanced sutured manifold (M, γ) detects which classes in H1(M) admit a taut depth one foliation such that the only compact leaves are the components of R(γ). The latter had been proved earlier by the first author under the extra assumption that H2(M) = 0. The main technical result is that we can obtain an extremal Spinc-structure s (i.e., one that is in a ‘corner’ of the support of SFH) via a nice and taut sutured manifold decomposition even when H2(M) ≠ 0, assuming the corresponding group SF H(M, γ, s) has nontrivial Euler characteristic