Embeddings of Homology Manifolds in Codimension

Among the many problems attendant to the discovery of exotic generalized manifolds [2, 3] is the \normal bundle" problem, that is, the classi cation of neighborhoods of generalized manifolds tamely embedded in generalized manifolds (with the disjoint disks property). In this paper we study the normal structure of tame embeddings of a closed generalized manifold X into topological manifolds V n+q , q 3. If the local index {(X) 6= 1, then the codimension is necessarily 3 (see, e.g., Proposition 5.4 below). The main result is an extension to ENR homology manifolds of the classi cation of neighborhoods of locally at embeddings of topological manifolds obtained by Rourke and Sanderson in [10]. We show that for q 3 and n+q 5, germs of tame manifold q-neighborhoods of X, or equivalently, controlled homeomorphism classes of (q 1)-spherical manifold approximate brations over X are in one-to-one correspondence with [X;BT opq ], where BT opq is the classifying space for stable topological q-microbundle pairs [10]. Manifold approximate brations over topological manifolds have been studied by Hughes, Taylor and Williams in [5]. Our approach is to reduce the study of q-neighborhoods of X to the classi cation of q-neighborhoods of a (stable) regular neighborhood of X in euclidean space; this is done using the splitting theorem for manifold approximate brations proved in section 4. As applications, we obtain an analogue of Browder's theorem on smoothings and triangulations of Poincar e embeddings (Theorem 5.1) and of the Casson-Hae iger-Sullivan-Wall embedding theorem (Corollary 5.2) for generalized manifolds. These were also obtained independently by Johnston in [6].