Homology manifold bordism

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact ANR homo logy manifolds of dimension greater than or equal to 6 is used to obtain tr ansversality, splitting and bordism results for homology manifolds, general izing previous work of Johnston. First, we establish homology manifold transversality for submanifolds of di mension greater than or equal to 7: if f : M --> P is a map from an m-dimen sional homology manifold M to a space P, and Q subset of P is a subspace wi th a topological q-block bundle neighborhood, and m-q greater than or equal to 7, then f is homology manifold s-cobordant to a map which is transverse to Q, with f(-1) (Q) subset of M an (m - q)-dimensional homology submanifo ld. Second, we obtain a codimension q splitting obstruction s(Q) (f) is an elem ent of LSm-q(Phi) in the Wall LS-group for a simple homotopy equivalence f : M --> P from an m-dimensional homology manifold M to an m-dimensional Poi ncare space P with a codimension q Poincare subspace Q subset of P with a t opological normal bundle, such that sQ (f) =0 if(and for m - q greater than or equal to 7 only if) f splits at Q up to homology manifold s-cobordism. Third, we obtain the multiplicative structure of the homology manifold bord ism groups Omega(*)(H) congruent to Omega(*)(TOP) [L-0(Z)].