The Borel Conjecture for manifolds with boundary

TL;DR

This paper develops a homological criterion for realizing certain closed manifolds as boundaries of compact aspherical manifolds, using surgery theory, with applications to manifolds with abelian fundamental groups.

Contribution

It introduces a new homological approach to characterize boundaries of aspherical manifolds with boundary, extending the theory in the context of abelian fundamental groups.

Findings

Established a homological criterion for boundary realization.

Constructed Poincaré pairs and applied surgery theory.

Illustrated results specifically for abelian fundamental groups.

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 Poincar\'e 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.