The simplicial volume of mapping tori of 3-manifolds

Abstract

We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume. In particular, we extend our computations to mapping tori of certain connected sums in higher dimensions. Our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four. Moreover, we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume. As a group theoretic consequence, we derive an alternative proof of the fact that the fundamental group G of a mapping torus of a 3-manifold M is Gromov hyperbolic if and only if M is virtually a connected sum #S 2 × S 1 and G does not contain Z 2 .

The spectrum of simplicial volume

“…Or represent the same class in H 4 b (G × ; R) by Proposition 3.6. Hence, , α× β = α 1 • β 1 .Moreover by the claim we have that3 2 • ∞ ≤ 1 and by duality we conclude that3 2 …”

“…More general surface bundles over surfaces do not yield lower estimates [2,43]. Also the recent computations/estimates for mapping tori in dimension 4 [3] do not produce improved gap bounds.…”

“…Proof of Claim 7. 3 We will follow the outline of Bucher's proof [4,Proposition 7], quoting parts of the proof verbatim. Let g 0 , .…”

Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume

“…The connection to the simplicial volume suggested by Theorem 1.3 is apparent: For the above fibration, M > 0 if and only if F and B are hyperbolic surfaces [12,Corollary 1.3]. Also, this definition should be absolutely compatible with maps of non-zero degree.…”

Length functions on mapping class groups and simplicial volumes of mapping tori

“…On the contrary, it is shown in [BN20] that the simplicial volume of any 4-dimensional closed orientable manifold fibering over the circle vanishes. This already shows that, if M is a closed orientable 3-manifold and ϕ is an element of MCG(M ) which may be represented by a homeomorphism, then FV(ϕ) = 0.…”