Let f : Mn→ Xn be a
homotopy equivalence between two closed, differentiable (of class C∞) n-manifolds
such that f induces, from the stable normal bundle Vk(Xn) of Xn in the
(n + k)-sphere Sn+k, a bundle over Mn that is equivalent to the stable normal
bundle Vk(Mn) of Mn in Sn+k. Then it is found that the disjoint union Xn∪ Mn
bounds a differentiable (n + 1)-manifold Wn+1 with a retraction r : Wn+1→ Xn
such that the restriction r∣Mn is equal to the given homotopy equivalence f.
Furthermore, let n = 2q − 1 or 2q, where q ≧ 3, and suppose that Xn is simply
connected if n = 2q − 1, and that Xn is 2-connected if n = 2q. Then, if the
restrictions of the bundles r∗Vk(Xn) and Vk(Wn+1) to the (q − 1)-skeleton of Wn+1
are equivalent, where r∗Vk(Xn) is the bundle induced by r : Wn+1→ Xn
and Vk(Wn+1) is the stable normal bundle of Wn+1 in Sn+k, then Mn
and Xn are diffeomorphic up to a point. In particular, Mn and Xn are
homeomorphic.
