Classification of Simplicial Triangulations of Topological Manifolds

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 3-spheres. 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