On the LS-category of homomorphism of almost nilpotent groups (Q6142718)

The LS-category \(\mbox{cat}(\phi)\) for a group homomorphism \(\phi : \Gamma \longrightarrow \land \) is the LS-category \(\mbox{cat}(f)\), where the map \(f : B\Gamma \longrightarrow B\land \) is induced by the homomorphism \(\phi\) on fundamental groups. The cohomological dimension of \(\mbox{cd}(\phi)\) is the maximum of \(k\) such that there exists a \(\mathbb Z\land\)-module \(M\) with a non-zero induced homomorphism \( H^k(\land;M) \longrightarrow H^k(\Gamma;M)\). In this paper, the authors prove that \(\mbox{cat}(\phi)=\mbox{cd}(\phi)\) for finitely generated, torsion-free almost nilpotent groups. Furthermore they prove: Let \(f : M^m \longrightarrow N^n\) be a map of infra-nilmanifolds that induces epimorphisms \(\phi : \pi_1(M) \longrightarrow \pi_1(N)\). Then \(\mbox{cat}(\phi)=\mbox{cd}(\phi)=n\). Here, an infra-nilmanifold means a closed manifold diffeomorphic to the orbit space \(G/\Gamma\) of a simply connected nilpotent Lie group \(G\) where the discrete torsion-free subgroup \(\Gamma\) acts on the semidirect product \(G \rtimes K\) where \(K\) is a maximal subgroup of \(\mbox{Aut}(G)\).