Vol. 26, No. 1, 1968

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Vol. 315: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
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