Abstract
Given a reducible 3-manifold M with an aspherical summand in its prime decomposition and a homeomorphism \(f:M\rightarrow M\), we construct a map of degree one from a finite cover of \(M\rtimes _f S^1\) to a mapping torus of a certain aspherical 3-manifold. We deduce that \(M\rtimes _f S^1\) 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 \((\#_n T^2\rtimes S^1)\#(\#_mS^2\times S^1)\). 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.
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Acknowledgements
The author is supported by the Swiss NSF, under grant FNS200021_169685. Part of this work was done during author’s visit at IHÉS in 2018. The author would like to thank IHÉS for providing a stimulating working environment, and especially Misha Gromov and Fanny Kassel for their hospitality.
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Neofytidis, C. Virtual Betti numbers of mapping tori of 3-manifolds. Math. Z. 296, 1691–1700 (2020). https://doi.org/10.1007/s00209-020-02485-w
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DOI: https://doi.org/10.1007/s00209-020-02485-w