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The Borel Conjecture for manifolds with boundary

James F. Davis, J. A. Hillman

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.

The Borel Conjecture for 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.
Paper Structure (7 sections, 17 theorems, 21 equations)

This paper contains 7 sections, 17 theorems, 21 equations.

Table of Contents

  1. Poincaré complexes, Poincaré pairs, and Poincaré duality groups
  2. Poincaré duality groups with boundary
  3. The Borel Uniqueness Conjecture for compact aspherical manifolds with boundary
  4. Homeomorphic boundaries
  5. Observations about the BUC
  6. The Borel Existence Conjecture for compact aspherical manifolds with boundary
  7. The classification of compact, aspherical manifolds with abelian fundamental group

Key Result

Theorem A

Let $M$ be a compact aspherical $n$-manifold with boundary, with fundamental group $\pi$, and with dimension $n$.

Theorems & Definitions (39)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Lemma 5
  • proof
  • ...and 29 more