The relationship between homology and topological manifolds via homology transversality

Abstract

In this paper we show that every polyhedral homology n-manifold, n > 6 (n > 5 if 8 M = g or if 8M is a topological manifold), is canonically simple homotopy equivalent to a topological n-manifold. This is accomplished by first observing in Section 3 that any two homology manifolds PL embedded in a PL manifold can be ambient isotoped to be in transverse position. Then, using the work of N. Levitt and J. Morgan [6], as refined by G. Brumfiel and J. Morgan [1], we show in Section 4 that the Spivak normal fiber space of any homology manifold has a canonical topological reduction. Finally, in Section 5 we show that the resulting topological surgery problem has zero surgery obstruction, thus showing that any homology n-manifold M with n > 6 (n > 5 if 8M =g) is simple homotopy equivalent to a topological manifold. We further show that if M is already a topological manifold, then the homotopy equivalence is homotopic to a homeomorphism. In Section 6 we define natural maps Gq: BH(q)-~BG(q) for q>3, and in Section 7 we define maps 0q: BH(q)-+BTOP(q) for q>3, such that