Four-manifolds of large negative deficiency

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.