Homeomorphisms between homotopy manifolds and their resolutions

Abstract

A homotopy n-manifold without boundary is a polyhedron M (i. e., a topological space along with a family of compatible triangulations by locally finite simplicial complexes) such that, for any triangulation in the piecewise linear (p. I.) structure of M, the link of each/-simplex (0 < i < n) has the homotopy type of the sphere S"-i-1. More generally, a homotopy n-manifold is a polyhedron such that the link of each /-simplex in any triangulation is homotopically an (n i 1) sphere or ball, and in which 0M the union of all simplexes with links which are homotopically balls is itself a homotopy ( n 1)-manifold without boundary. We note that i fM is a homotopy manifold then 0M is a well-defined subpolyhedron of M. Also, the question of whether a polyhedron M is a homotopy manifold is completely determined by a single triangulation of M (by Lemma LK 5 of [8]).