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 ⩾3. 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.