# The simplicial volume of mapping tori of 3-manifolds

## TL;DR

This paper proves that all mapping tori of closed 3-manifolds have zero simplicial volume, extending the result to higher dimensions and providing new techniques for analyzing self-homeomorphisms of reducible 3-manifolds.

## Contribution

It introduces a novel method for understanding self-homeomorphisms of connected sums and computes the simplicial volume of mapping tori, completing the classification in dimension four.

## Key findings

- All mapping tori of closed 3-manifolds have zero simplicial volume.
- The techniques extend to certain higher-dimensional connected sums.
- Dimension four is unique in having all mapping tori with vanishing simplicial volume.

## 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 $\# S^2\times S^1$ and $G$ does not contain $\mathbb{Z}^2$.