  We classify closed, topological spin$^+$ 4-manifolds with fundamental group
$\pi$ of cohomological dimension $\leq 3$ (up to s-cobordism), after
stabilization by connected sum with at most $b_3(\pi)$ copies of $S^2\times
S^2$. In general we must also assume that $\pi$ also satisfies certain K-theory
and assembly map conditions. Examples for which these conditions hold include
the torsion-free fundamental groups of 3-manifolds and all right-angled Artin
groups whose defining graphs have no 4-cliques.
