Aspherical manifolds with boundary

Abstract

We undertake a systematic investigation of compact aspherical manifolds with boundary; motivated by the plethora of examples in the bounded case and by the beauty of the theory in the closed case. Our main theorems give a homological criterion for when a closed manifold, together with maps from the fundamental groups of its components to a fixed group, can be realized as the boundary of a compact aspherical manifold. This is done in two steps: we first produce a Poincaré pair and then apply surgery theory to obtain a manifold. We illustrate this in the case of abelian fundamental group. The results of this paper will be applied in a sequel where we classify compact aspherical 4-manifolds with elementary amenable fundamental group.

New title. Added Conjecture B, which gives a homological criterion for the homotopy type of a compact aspherical manifold (which is the main theme of the paper). Added details for the proof of Theorem C. Generalized Theorem E from homotopy tori to homotopy K(G,1)s with G a duality group, for example, a homotopy figure eight; 33 pages