Transversality for Homology Manifolds

Transversality phenomena are studied for homology manifolds. For homology manifolds X, Y and Z, with Z embedded in Y with a neighborhood ν(Z) which has 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(ν(Z)) has a bundle structure given by f∗ν(Z) and f induces the bundle map. In the case where the range is a manifold an arbitrary map is s-cobordant to a transverse map if the submanifold is codimension one and (π, π) or codimension greater than two. Appropriate homology manifold versions of related 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.