Aspherical 4-manifolds with elementary amenable fundamental group

TL;DR

This paper classifies the fundamental groups of certain 4-manifolds, establishes topological rigidity for them, and characterizes their boundary 3-manifolds, extending previous results and providing a detailed structural understanding.

Contribution

It provides a classification of elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and proves topological rigidity for these manifolds.

Findings

Fundamental groups are either polycyclic or solvable Baumslag-Solitar.

Such manifolds satisfy topological rigidity: homotopy equivalences rel boundary are homotopic to homeomorphisms.

Boundary 3-manifolds are classified as arising from these 4-manifolds, generalizing prior results.

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.