Virtual Betti numbers of mapping tori of 3-manifolds (Q2205619)

The author considers mapping tori \(M\rtimes_f S^1\) of homeomorphisms \(f\) of reducible \(3\)-manifolds \(M\). Under the assumption that \(M\) contains an aspherical summand in its prime decomposition, the author starts by constructing a degree-one map from a finite cover of \(M\) to an aspherical \(3\)-manifold. This result, together with a description of the isotopy classes of homeomorphisms of reducible \(3\)-manifolds, is then exploited to build a degree-one map from a finite cover of \(M\rtimes_f S^1\) to a mapping torus of a homemomorphism of an aspherical \(3\)-manifold. As a consequence, mapping tori of homeomorphisms of reducible \(3\)-manifolds \(M\) have \textit{infinite virtual first Betti number} provided that the prime decomposition of \(M\) (1) contains an aspherical summand and (2) does not contain only virtual torus-bundles. Recall that a manifold has infinite virtual first Betti number if the set of first Betti numbers of its finite covers is unbounded. This consequence was previously established by \textit{Y. Ni} [Sci. China, Math. 60, No. 9, 1591--1598 (2017; Zbl 1385.57029)] under the weaker hypothesis that \(M\) is not finitely covered by \(S^2\times S^1\), exploiting algebraic properties of the fundamental groups of the manifolds.