Topological 4-manifolds with 4-dimensional fundamental group
Daniel Kasprowski, Markus Land

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 be a group satisfying the Farrell-Jones conjecture and assume that is a 4-dimensional Poincar\'e duality space. We consider topological, closed, connected manifolds with fundamental group whose canonical map to 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 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.
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