Vol. 26, No. 1, 1968

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Almost diffeomorphisms of manifolds

Rodolfo DeSapio

Vol. 26 (1968), No. 1, 47–56

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 V k(Xn) of Xn in the (n + k)-sphere Sn+k, a bundle over Mn that is equivalent to the stable normal bundle V k(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 rMn 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 rV k(Xn) and V k(Wn+1) to the (q 1)-skeleton of Wn+1 are equivalent, where rV k(Xn) is the bundle induced by r : Wn+1 Xn and V k(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.

Mathematical Subject Classification
Primary: 57.10
Received: 7 March 1967
Published: 1 July 1968
Rodolfo DeSapio