Tangential homotopy equivalences

Two (topological) manifolds M and are called tangentially homotopy equivalent if there exists a homotopy equivalence f: (N, 3 N ) ~ (M, OM) such that f*(~'M) is stably equivalent to ~'N. Let O(m) denote the set of homeomorphisms types of manifolds which are tangentially homotopy equivalent to M. In this paper we study O(M). In particular we give estimates of its size for suitable classes of manifolds. Given any set S we use IS1 to denote its cardinality.