The topology of homology manifolds: Nice properties of some very singular spaces

A homology manifold is a generalization of a manifold characterized by local Poincare duality. In 1983, Quinn defined an obstruction to resolution for homology manifolds: an integer congruent to 1 mod 8 which corresponds to a "0th Pontrjagin class." In 1993, Bryant, Ferry, Mio and Weinberger produced the first examples of homology manifolds which are not resolvable by manifolds. These non-resolvable homology manifolds have no manifold points whatsoever. Since they have very bad local structures, they do not have nice geometric properties of manifolds such as transversality.

Nonetheless, the algebraic structure of homology manifolds is sufficient to prove an analogue of the Browder-Novikov-Sullivan-Wall surgery exact sequence, which is used to classify the topological manifold structures within a given homotopy type. For homology manifolds the Bryant-Ferry-Mio-Weinberger surgery exact sequence classifies homology manifolds within a given homotopy type up to s-cobordism. Various bordism, partial transversality,and embedding results up to s-cobordism are proven using the homology manifold surgery exact sequence. However, homogeneity and the s-cobordism conjecture for homology manifolds with the disjoint disks property remain open.