Classifications of simplicial triangulations of topological manifolds

Abstract

In this note we announce theorems which classify simplicial (not necessarily combinatorial) triangulations of a given topological «-manifold M, n > 7 (> 6 if dM = 0 ) , in terms of homotopy classes of lifts of the classifying map r: M —• BTOP for the stable topological tangent bundle of M to a classifying space BTRIn which we introduce below. The (homotopic) fiber of the natural map ƒ: BTRIn —• BTOP is described in terms of certain groups of PL homology 3spheres. We also give necessary and sufficient conditions for a closed topological «-manifold M, n> 6, to possess a simplicial triangulation. The proofs of these results incorporate recent geometric results of F. Ancel and J. Cannon [1] , J. Cannon [2], R. D. Edwards [4] , and D. Galewski and R. Stern [5]. In [8], R. Kirby and L. Siebenmann show that in each dimension greater than four there exist closed topological manifolds which admit no piecewise linear manifold structure and hence cannot be triangulated as a combinatorial manifold. Also, R. D. Edwards [3] has recently shown that the double suspension of the Mazer homology 3-sphere is homeomorphic to S, thus showing that a simplicial triangulation of a topological manifold need not be combinatorial. But it is still unknown whether or not every topological manifold can be triangulated as a simplical complex. Our classification theorems for simplicial triangulations on a given topological manifold take the following forms: Let BTOP denote the classifying space for stable topological block bundles.