The Characterization of Topological Manifolds of Dimension n > 5

That rich, unkempt world of wild and tame topology, born in the minds of Antoine and Alexander, recalled from obscurity by Fox, Artin, and Moise, and brought to full bloom by Bing, has spawned a conjecture on the nature of the topological manifold having as one of its minor corollaries the famous double suspension theorem for homology spheres. F. Quinn in the Saturday morning topology seminar of this congress expressed confidence that he has the right conceptual and technical framework to complete the final step in its proof. Whatever the result after Quinn has had opportunity to verify his intuitions, the result is at the very least almost true; and we wish to discuss it. As is often the case, much of the visualization and example which gave the conjecture birth will surely disappear in the powerful application of engulfing, local surgery, etc., which should constitute its final proof. And so, for those of us who have always savored the interplay among point set topology, taming theory, decomposition space theory, and other visual aspects of geometric topology, we record here the milieu in which the conjecture became reasonable and the pressures leading to its formulation. But first we summarize the conjecture itself and its most recent history. In the early spring of 1977 we conjectured,