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.