This method consists in solving first a lifting problem and then dealing with a secondary obstruction. Unfortunately, in concrete cases, even the first step can be quite delicate. The relation between connecting manifolds and smoothing problems is the following. In this way one can identify non-smoothable homotopy types and one does recover classical examples of Smith, Kervaire and also new such examples. Fall break at Georgia Tech.
Homology manifolds are generalizations of manifolds.
In 1978, Cannon conjectured that all homology manifolds were resolvable by manifolds. That is, that every homology manifold could be paired with a true manifold, to which it was locally homotopy equivalent. In 1984, Quinn found an integer which captured the obstruction to this resolution. In 1993, Bryant, Ferry, Mio and Weinberger discovered the first examples of non-resolvable homology manifolds. These spaces have no manifold points whatsoever, yet they have many of the geometric properties of manifolds.