For every $N > 0$ there exists a group of deficiency less than ${-}N$ that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold of the $n$-sphere, $n > 4$, if and only if it is the fundamental group of a homology $n$-sphere. There exist fundamental groups of homology $n$-spheres, $n > 4$, that cannot arise as the fundamental group of the complement of a contractible submanifold of the 4-sphere.