Deforming homotopy equivalences to homeomorphisms in aspherical manifolds

Abstract

Aspherical manifolds are closed manifolds which are A"(w, l)'s. They play a significant role in many branches of mathematics. This paper constructs "model" aspherical manifolds for various classes of IT and investigates and surveys their groups of homeomorphisms. It is not known whether aspherical manifolds having the same fundamental groups as our model manifolds can topologically differ from them. If m contains a normal abelian subgroup then the model aspherical manifolds are special instances of injective Seifert fiber spaces. The group of (singular) fiber preserving homeomorphisms of Seifert fiber spaces are characterized and criteria obtained so that each self-homotopy equivalence may be deformed to such a homeomorphism. Many of our model aspherical manifolds satisfy these criteria. Other applications are given to group theory, differential geometry, complex manifolds as well as topology. We have also included a list of unsolved problems.