A criterion for homeomorphism between closed Haken manifolds

Abstract

In this paper we consider two connected closed Haken manifolds denoted by $M^3$ and $N^3$, with the same Gromov simplicial volume. We give a simple homological criterion to decide when a given map $f:M^3\to N^3$ between $M^3$ and $N^3$ can be changed by a homotopy to a homeomorphism. We then give a convenient process for constructing maps between $M^3$ and $N^3$ satisfying the homological hypothesis of the map $f$.