The boundary of a contractible compact manifold must have the same integral homology as a sphere, but may have non-trivial fundamental group.  Examples are easy to come by, and were studied by Kervaire in 1969, among others.  I will discuss the companion question of understanding the automorphisms (homeomorphisms or diffeomorphisms) of contractible compact manifolds, and the extent to which their automorphism groups differ from that of the standard disk of the same dimension.  Based on joint work with Oscar Randal-Williams (arXiv: 2308.15607)