Sutured Floer homology, fibrations, and taut depth one foliations

Abstract:

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 $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,\gamma )$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(\gamma )$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\operatorname {Spin}^c$-structure $\mathfrak {s}$ (i.e., one that is in a ‘corner’ of the support of $SFH$) via a nice and taut sutured manifold decomposition even when $H_2(M) \neq 0$, assuming the corresponding group $SFH(M,\gamma ,\mathfrak {s})$ has non-trivial Euler characteristic.