# Commentarii Mathematici Helvetici

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**Volume 85, Issue 1, 2010, pp. 1–37**

**DOI: 10.4171/CMH/186**

Published online: 2009-12-23

Topological rigidity and Gromov simplicial volume

Pierre Derbez^{[1]}(1) Université de Provence, Marseille, France

A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps.

To this purpose we introduce an invariant `τ`(`N`) = (Vol(`N`),||N||), where ||N|| denotes the Gromov simplicial volume of
`N` and Vol(`N`) is a 2-dimensional simplicial volume which measures the volume of the base 2-orbifolds of the Seifert pieces of `N`.

After studying the behavior of `τ`(`N`) under the action of non-zero degree maps, we prove that if `M` and `N` are closed Haken manifolds
such that ||`M`|| = |deg(`f`)| ||`N`|| and Vol(`M`) = Vol(`N`) then any non-zero degree
map `f`: `M` → `N` is homotopic to a covering map.
As a corollary we prove that if `M` and `N` are closed Haken manifolds such that `τ`(`N`) is sufficiently close to `τ`(`M`) then any degree one map
`f`: `M` → `N` is homotopic to a homeomorphism.

*Keywords: *Haken manifold, Seifert fibered space, hyperbolic 3-manifold, Gromov simplicial volume, non-zero degree maps, Dehn filling, subgroup separability

Derbez Pierre: Topological rigidity and Gromov simplicial volume. *Comment. Math. Helv.* 85 (2010), 1-37. doi: 10.4171/CMH/186