Virtual Betti numbers of mapping tori of 3-manifolds

Abstract

Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f:M\to M$, we construct a map of degree one from a finite cover of $M{⋊}_f{S}^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M{⋊}_f{S}^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except when $M$ is virtually $({\#}_n{T}^2⋊S^1)\#({\#}_m{S}^2\times S^1)$. Li-Ni's conjecture was recently confirmed by Ni with a group theoretic result, namely, by showing that there exists a ${\pi }_1$-surjection from a finite cover of any mapping torus of a reducible $3$-manifold to a certain mapping torus of ${\#}_m{S}^2\times S^1$ and using the fact that free-by-cyclic groups are large when the free group is generated by more than one element.

11 pages; v2: typos fixed, to appear in Mathematische Zeitschrift