  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.
