In this paper we consider two connected closed Haken manifolds denoted by and , with the same Gromov simplicial volume. We give a simple homological criterion to decide when a given map between and can be changed by a homotopy to a homeomorphism. We then give a convenient process for constructing maps between and satisfying the homological hypothesis of the map .
"A criterion for homeomorphism between closed Haken manifolds." Algebr. Geom. Topol. 3 (1) 335 - 398, 2003. https://doi.org/10.2140/agt.2003.3.335