Taut sutured manifolds and twisted homology (Q395053)

The authors give a necessary and sufficient criterion for a sutured manifold \((M,\gamma)\) to be taut in terms of the twisted homology of the pair \((M,R_-)\).NEWLINENEWLINEA sutured manifold \((M,\gamma)\) is a compact, connected, oriented 3-manifold \(M\) together with a set of disjoint annuli \(\gamma\) on \(\partial M\), which turn \(M\) naturally into a cobordism between oriented surfaces \(R_-=R_-(\gamma)\) and \(R_+=R_+(\gamma)\) with boundary. This notion was introduced by \textit{D. Gabai} [J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)]. We say that a sutured manifold \((M,\gamma)\) is balanced if \(\chi(R_+)=\chi(R_-)\), where \(\chi\) denotes the Euler characteristic.NEWLINENEWLINEFrom the authors' introduction: ``Given a surface \(S\) with connected components \(S_1\cup\cdots\cup S_\ell\) we define its complexity to be \(\displaystyle{\chi_-(S)=\sum_{i=1}^{\ell}\max\{-\chi(S_i),0\}}\). Following Gabai [loc. cit.]we say that a balanced sutured manifold \((M,\gamma)\) is taut if \(M\) is irreducible and if \(R_-\) and \(R_+\) have minimal complexity among all surfaces representing the homology class \([R_-]=[R_+]\in H_2(M,\gamma;\mathbb Z)\).NEWLINENEWLINEGiven a representation \(\alpha : \pi_1(M)\to \text{GL}(k,\mathbb F)\) over a field \(\mathbb F\) we can consider the twisted homology groups \(H^{\alpha}_*(M,R_-;\mathbb F^k)\). In this paper we give a necessary and sufficient criterion for a balanced sutured manifold \((M, \gamma)\) to be taut in terms of the twisted homology of the pair \((M,R_-)\). More precisely we wil prove the following theorem:NEWLINENEWLINE\noindent Theorem 1.1. Let \((M,\gamma)\) be an irreducible balanced sutured manifold with \(M\neq S^1\times D^2\) and \(M\neq D^3\). Then \((M,\gamma)\) is taut if and only if \(H^{\alpha}_1(M,R_-;\mathbb C^k)=0\) for some unitary representation \(\alpha:\pi_1(M)\to U(k)\).''NEWLINENEWLINEThis theorem is obtained from the following two theorems. The ``if'' direction can be proved using classical methods. It is obtained from the next theorem (Corollary 3.2).NEWLINENEWLINE\noindent Theorem. Let \((M,\gamma)\) be an irreducible balanced sutured manifold and let \(\mathbb F\) be a field with involution. Assume there exists a unitary representation \(\alpha : \pi_1(M)\to \text{GL}(k,\mathbb F)\) (i.e., a representation such that \(\alpha(g^{-1})=\overline{\alpha(g)}^t\) for all \(g\in \pi_1(M)\)) such that \(H_1(M,R_-;\mathbb F^k)=0\). Then \((M,\gamma)\) is taut.NEWLINENEWLINEThe following theorem (Theorem 4.1) is a slight strengthening of the ``only if '' part of Theorem 1.1.NEWLINENEWLINE\noindent Theorem. Let \((M,\gamma)\) be a taut sutured manifold with \(M\neq S^1\times D^2\) and \(M\neq D^3\). Then there exists a unitary representation \(\alpha:\pi_1(M)\to U(k)\), which factors through a finite group such that \(H_*(M,R_-;\mathbb C^k)=H_*(M,R_+;\mathbb C^k)=0\).NEWLINENEWLINEIn order to prove this theorem, the authors use the recent works of \textit{I. Agol} [J. Topol. 1, No. 2, 269--284 (2008; Zbl 1148.57023)], Liu and Przytycki-Wise and Wise.