Topological 4-manifolds with 4-dimensional fundamental group

TL;DR

This paper establishes conditions under which topological 4-manifolds with a specific fundamental group are classified up to s-cobordism and homeomorphism, linking algebraic invariants to topological equivalences.

Contribution

It provides new classification criteria for 4-manifolds with fundamental groups satisfying the Farrell-Jones conjecture, extending rigidity results between aspherical and simply connected cases.

Findings

Two such manifolds are s-cobordant if their intersection forms are isometric and Kirby-Siebenmann invariants match.

If the group is good in Freedman's sense, they are homeomorphic if homotopy equivalent and invariants agree.

The results bridge the gap between aspherical and simply connected 4-manifold classifications.

Abstract

Let $\pi$ be a group satisfying the Farrell-Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1 and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby-Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby--Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results.