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×R~(n-k) singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R~(n+1) with smooth boundary ?K for which the resulting mean curvature flow has a shrinking Sm×Rn-m singularity.
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