Vol. 38, No. 2, 1971

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A proof of the most general polyhedral Schoenflies conjecture possible

Leslie C. Glaser

Vol. 38 (1971), No. 2, 401–417

Here, it is shown that if Mn is an n-manifold triangulated as a locally finite simplicial complex and Nk is a closed subcomplex of int Mn that is also a topological k-manifold, then Nk is topologically locally flat in Mn provided nk2 and each of Nk and Mn is a simplicial homotopy manifold. This result not only generalizes all known results to date, but also either includes the most general case, where no further assumptions on the triangulations are made, or the general case is false in a very strong sense. That is, if some triangulated topological n-manifold is not a simplicial homotopy n-manifold, then there exist, for some m, a triangulated m-sphere Σ and PL(m 1)-and (m + 1)-spheres S and 𝒮, respectively, such that Σ is a subcomplex of 𝒮,S is a subcomplex of Σ,𝒮− Σ = U V , where U is homeomorphic to Em+1, but π1(V )0, and S bounds a PL m-ball B in Σ, but π1B)0. The main result is obtained by noting some results related to double suspensions of homotopy 3-and 4-spheres and showing that each open simplex of such a triangulation, as above, is topologically flat in the given manifold.

Mathematical Subject Classification
Primary: 57C45
Received: 5 January 1970
Published: 1 August 1971
Leslie C. Glaser