Topological rigidity and Gromov simplicial volume

Abstract

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)=({\rm Vol}(N),\|N\|)$ where $\|N\|$ denotes the Gromov simplicial volume of $N$ and ${\rm Vol}(N)$ 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 nonzero degree maps action, we prove that if $M$ and $N$ are closed Haken manifolds such that $\|M\|=\abs{{\rm deg}(f)}\|N\|$ and ${\rm Vol}(M)={\rm Vol}(N)$ then any non-zero degree map $f\co M\to N$ is homotopic to a covering map. This extends a result of S. Wang in \cite{W1} for maps of nonzero degree from $M$ to itself. 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 $f\co M\to N$ is homotopic to a homeomorphism.