  Consider the mean curvature flow of an (n+1)-dimensional, compact, mean
convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We
prove that elements of the m-th homotopy group of the complementary region can
die only if there is a shrinking S^k x R^(n-k) singularity for some k less than
or equal to m. We also prove that for each m from 1 to n, there is a nonempty
open set of compact, mean convex regions K in R^(n+1) with smooth boundary for
which the resulting mean curvature flow has a shrinking S^m x R^(n-m)
singularity.
