Let S be an (n — l)-sphere smoothly embedded in a closed, orientable, smooth n-manifold M, and let the embedding be null-homotopic. We show that, if S does not bound a ball, then M is a rational homology sphere, the fundamental groups of both components of M \ S are finite, and at least one of them is trivial.

The following theorem describes what can happen; there are examples in every dimension to show that this is (more or less) the best possible. The only qualification is that it is perhaps possible to show that both X and Y must be simply connected; all of the examples constructed at the end of this article have this property. THEOREM 1. Suppose that t : S n – 1 → Mn is a null-homotopic smooth embedding. Then either S bounds a homotopy ball on one side, or the following statements hold:

• (i) M is a rational homology sphere, and therefore X and Y are as well.

• (ii) The fundamental groups of both X and Y are finite, and at least one of them is trivial.

For n > 4, if S bounds a homotopy ball then it bounds a (smooth) ball, while if n = 4 it bounds a topological bal

The basic ingredient in the proof is the well-known principle that a manifold admitting a map from a sphere of nonzero degree must be a rational homology sphere.