Transversality for homology manifolds

Transversality phenomena are studied for homology manifolds. For homology m anifolds X, Y and Z, with Z embedded in Y with a neighborhood nu(Z) which h as a given bundle structure, we define a map f:X --> Y to be transverse to Z, if f(-1)(Z) = Z' is a homology manifold, the neighborhood f(-1)(nu(Z)) h as a bundle structure given by f*nu(Z) and f induces the bundle map. In the case where the range is a manifold an arbitrary map is s-cobordant to a tr ansverse map if the submanifold is codimension one and (pi, pi) or codimens ion greater than or equal to 3. Appropriate homology manifold versions of r elated splitting and embedding theorems are proved for homology manifolds. As a group, bordism of high dimensional homology manifolds has one copy of the bordism of topological manifolds for each possible index. (C) 1999 Else vier Science Ltd. All rights reserved.