On the Hauptvermutung for manifolds

Abstract

The is the conjecture that homeomorphic (finite) simplicial complexes have isomorphic subdivisions, i.e. homeomorphic implies piecewise linearly homeomorphic. I t was formulated in the first decade of this century and seems to have been inspired by the question of the topological invariance of the Betti and torsion numbers of a finite simplicial complex. The Hauptvermutung is known to be true for simplicial complexes of dimension 4 (Milnor, 1961). The Milnor examples, K and L, have two notable properties: (i) K and L are not manifolds, (ii) K and L are not locally isomorphic. Thus it is natural to restrict the Hauptvermutung to the class of piecewise linear w-manifolds, simplicial complexes where each point has a neighborhood which is piecewise linearly homeomorphic to Euclidean space R or Euclidean half space R\. We assume that Hz(M, Z) has no 2-torsion.