Virtual Betti numbers of mapping tori of 3-manifolds
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Mathematische Zeitschrift
Virtual Betti numbers of mapping tori of 3-manifolds Christoforos Neofytidis1 Received: 19 October 2018 / Accepted: 26 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a reducible 3-manifold M with an aspherical summand in its prime decomposition and a homeomorphism f : M → 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 × S 1 has virtually infinite first Betti number, except when M is virtually (#n T 2 S 1 )#(#m S 2 × S 1 ). Li-Ni’s conjecture was recently confirmed by Ni with a group theoretic result, namely, by showing that there exists a π1 -surjection from a finite cover of any mapping torus of a reducible 3-manifold to a certain mapping torus of #m S 2 × S 1 and using the fact that free-by-cyclic groups are large when the free group is generated by more than one element. Keywords Virtual Betti numbers · Mapping tori of reducible 3-manifolds · Degree one maps Mathematics Subject Classification 57M05 · 57M10 · 57M50 · 55M25 · 57N37
1 Introduction The virtual first Betti number of a manifold M is defined to be vb1 (M) = sup b1 (M) | M is a finite cover of M (where b1 denotes the first Betti number) and takes values in N0 ∪ {∞}. This notion arises naturally in geometric topology and it is often difficult to compute. A recent prominent example is given by the resolution of the Virtual Haken Conjecture [1] which implies that vb1 = ∞ for hyperbolic 3-manifolds, and therefore completes the picture for the values of vb1 in dimension three. Li and Ni [10] used this picture to compute vb1 for mapping tori of prime 3-manifolds: Theorem 1.1 [10, Theorem 1.2] Let X = M f S 1 be a mapping torus of a closed prime 3-manifold M. Then vb1 (X ) is given as follows:
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Christoforos Neofytidis [email protected] Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
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C. Neofytidis
(1) If M is a spherical manifold, then vb1 (X ) = 1; (2) If M is S 1 × S 2 or finitely covered by T 2 S 1 , then vb1 (X ) ≤ 4; (3) In all other cases, vb1 (X ) = ∞. When the fiber M is reducible, then the monodromy f of the mapping torus M f S 1 is in general more complicated than when M is irreducible; see [11,12,15]. Li and Ni conjectured that almost always vb1 (M f S 1 ) = ∞ when M is reducible: Conjecture 1.2 [10, Conjecture 5.1] If M is a closed oriented reducible 3-manifold, then vb1 (M f S 1 ) = ∞, unless M is finitely covered by S 2 × S 1 . As pointed out in [10, Lemma 5.4] (see Lemma 2.2), if M is a closed reducible 3manifold which is not covered by S 2 × S 1 , then M is finitely covered by a connected sum M #(S 2 × S 1 )#(S 2 × S 1 ), for some closed 3-manifold M . Since free-by-cyclic gro
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