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Aspherical 4-manifolds with elementary amenable fundamental group

James F. Davis, J. A. Hillman

TL;DR

The paper classifies the elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary, showing they are restricted to polycyclic groups or solvable Baumslag–Solitar groups $BS(1,m)$. It develops a framework of peripheral systems and enhanced peripheral systems to encode boundary data, proves existence of realizations of prescribed boundary data via $PD_4$-pair and Borel existence results, and establishes a uniqueness theorem: two such 4-manifolds with equivalent enhanced peripheral systems are homeomorphic. The authors then provide a detailed classification: manifolds are determined up to homeomorphism by the ambient group, boundary 3-manifolds, and how the boundary embeds into the interior, with the boundary connected-sum decomposition into a prime boundary piece and a contractible summand. They also discuss concrete realizations and present examples illustrating limits of uniqueness and the role of boundary data, along with extensions and open questions about broader classes of groups and smooth structures. The work integrates 3-manifold theory, topological surgery, and low-dimensional group theory under the Farrell–Jones framework to yield a robust rigidity and realization theory for this class of 4-manifolds.

Abstract

We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold.

Aspherical 4-manifolds with elementary amenable fundamental group

TL;DR

The paper classifies the elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary, showing they are restricted to polycyclic groups or solvable Baumslag–Solitar groups . It develops a framework of peripheral systems and enhanced peripheral systems to encode boundary data, proves existence of realizations of prescribed boundary data via -pair and Borel existence results, and establishes a uniqueness theorem: two such 4-manifolds with equivalent enhanced peripheral systems are homeomorphic. The authors then provide a detailed classification: manifolds are determined up to homeomorphism by the ambient group, boundary 3-manifolds, and how the boundary embeds into the interior, with the boundary connected-sum decomposition into a prime boundary piece and a contractible summand. They also discuss concrete realizations and present examples illustrating limits of uniqueness and the role of boundary data, along with extensions and open questions about broader classes of groups and smooth structures. The work integrates 3-manifold theory, topological surgery, and low-dimensional group theory under the Farrell–Jones framework to yield a robust rigidity and realization theory for this class of 4-manifolds.

Abstract

We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold.
Paper Structure (8 sections, 18 theorems, 17 equations)

This paper contains 8 sections, 18 theorems, 17 equations.

Table of Contents

  1. Group theoretic notation and terminology
  2. General remarks on the peripheral systems of aspherical manifolds
  3. Fundamental group systems of aspherical 4-manifolds
  4. criteria for asphericity
  5. existence: realization of boundaries
  6. uniqueness: the role of the peripheral system
  7. Classification
  8. other good groups

Key Result

Theorem A

Let $M$ be a compact aspherical $4$-manifold with boundary $N$, and let $\mathop{\mathrm{inc}}\nolimits:N\to{M}$ be the inclusion. If $\pi=\pi_1M$ is elementary amenable, then one of the following conditions holds:

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Lemma 4
  • ...and 26 more