Proper Homotopy Nonrigidity of Open Contractible Manifolds

Stallings’ characterization of Euclidean space implies that the proper homotopy type of ${\mathbb{R}}^n$ is topologically rigid for $n\ge 5$. We show that this phenomenon is exceptional. For every even integer $N\ge 6$, there exists a proper homotopy type containing infinitely many pairwise nonhomeomorphic smooth open contractible $N$-manifolds. More generally, let $N=2d\ge 6$, and let $\pi$ be a finite superperfect group. If the reduced $(-1{)}^d$-eigenspace of the rational complex representation ring of $\pi$ is nonzero, then there exist infinitely many compact contractible smooth $N$-manifolds whose interiors are all properly homotopy equivalent but pairwise nonhomeomorphic. Their boundaries are homotopy equivalent integral homology $(N-1)$-spheres with fundamental group $\pi$, but are pairwise not topologically $h$-cobordant.