Intrinsic transversality structures

Abstract

This paper introduces the notion of an intrinsic transversality structure on a Poincaré duality space Xn. Such a space has an intrinsic transversality structure if the embedding of Xn into its regular neighborhood Wn+k in Euclidean space can be made "Poincare transverse" to a triangulation of Wn+k. This notion relates to earlier work concerning transversality structures on spherical fibrations, which are known to be essentially equivalent to topological bundle reductions. Thus, for n ≥ 5, a Poincare duality space Xn with a transversality structure on its Spivak normal fibration (i.e., with an "extrinsic" transversality structure) is, up to a surgery obstruction, realizable as a topological manifold. An intrinsic transversality structure, however, not only guarantees the existence of an extrinsic transversality structure but gives rise as well to a canonical solution of the resulting surgery problem. Thus, as our main result, an equivalence is obtained between intrinsic transversality structures and topological manifold structures. This yields a number of corollaries, among which the most important is a "local formula for the total surgery obstruction" which assembles this obstruction to the existence of a manifold structure on Xn from the local singularities of a realization of the simple homotopy type of Xn as a (non-manifold) simplicial complex. © 1987 by Pacific Journal of Mathematics.