A collaring theorem for codimension one manifolds

The chief result implies that an n-manifold S embedded in the interior of an (n + 1)-manifold M as a closed, separating subset is locally flatly embedded if the embedding is well behaved in a locally peripheral sense and if S has arbitrarily close neighborhoods Q such that the fundamental groups of appropriate components of Q \ S admit a uniform finite upper bound on the number of generators. Let M denote a compact (n + 1)-manifold, possibly with boundary, and let S c Int M denote a closed n-manifold such that M \ S has two components U and U'. The main theorem establishes that S has a collar in S u U if the fundamental group of the end of U is finitely generated and S is locally peripherally collared from U. As a corollary, S has a collar in S U U if S is locally flat except at the points of a Cantor set C c S standardly embedded in S and if U is an open collar (that is, U is homeomorphic to the product of (0, 1] with another closed n-manifold). Interest in the issue emerged from our constructions [DT 1, DT2] of (n + 1)manifolds M formed as a union of copies of S x [-1, 0] and S' x (0, 1], where S, S' are closed n-manifolds of different homotopy types and the level corresponding to S x 0 is locally flat modulo a Cantor set. In these constructions one observes that the Cantor set turns out to be wildly embedded in S x 0. Is wildness necessary? The result here gives an affirmative answer, for otherwise a bicollar on S x 0 would force S and S' to be homotopically equivalent. Influential in these developments was Kirby's prototype flatness result [K], which promises that an n-sphere S in Rn (n > 3) is bicollared if it is locally flat modulo a Cantor set that is twice tame-tame both in Rn and in S. Another more direct influence was the paper of Burgess [B], which seems to be the first to combine conditions on local peripheral structure, defined in the next paragraph, with global conditions near the codimension one submanifold, to derive conclusions about local flatness, in the case of [B], for embeddings in 3-manifolds. Let M be a connected (n + 1)-manifold, possibly with boundary, S c Int M Received by the editors June 18, 1992. 1991 Mathematics Subject Classification. Primary 57N45, 57N35; Secondary 57N40, 57N70.