A proof of the most general polyhedral Schoenflies conjecture possible

Leslie C. Glaser

Vol. 38 (1971), No. 2, 401–417

DOI: 10.2140/pjm.1971.38.401

Abstract

Here, it is shown that if Mⁿ is an n-manifold triangulated as a locally finite simplicial complex and N^k is a closed subcomplex of int Mⁿ that is also a topological k-manifold, then N^k is topologically locally flat in Mⁿ provided n⋅k≠2 and each of N^k 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 E^m+1, but π₁(V )≠0, and S bounds a PL m-ball B in Σ, but π₁(Σ − B)≠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

Milestones

Received: 5 January 1970

Published: 1 August 1971