Topological change in mean convex mean curvature flow

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 mth homotopy group of the complementary region can die only if there is a shrinking S (k) xR (n-k) singularity for some ka parts per thousand currency signm. We also prove that for each m with 1a parts per thousand currency signma parts per thousand currency signn, there is a nonempty open set of compact, mean convex regions K in R (n+1) with smooth boundary a,K for which the resulting mean curvature flow has a shrinking S (m) xR (n-m) singularity.