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On Stable Diffeomorphism of Exotic Spheres in the Metastable Range

Published online by Cambridge University Press:  20 November 2018

P. L. Antonelli
P. L. Antonelli*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey The University of Alberta, Edmonton, Alberta
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Let ⊖np+ 1 denote the subgroup of the Kervaire-Milnor group θn consisting of those n-spheres which imbed with trivial normal bundle in Euclidean (n + p + 1)-space, n < 2p. It is known that such imbeddings always exist [6], and that the normal bundle is independent of the imbedding [10]. Following [2], we write Ωn,p for the quotient θn/⊖np+ 1.

The order of Ωn,p after identifying each element with its inverse, is equal to the number of diffeomorphically distinct (orientation preserved) Σn × Sp [2; 5]. Indeed, Ωn,p is closely linked to the problem of determining the number of smooth structures α(n, p) on Sn × Sp. For instance, if Ωn,p = 0 then α(n, p) equals the order of θn+p [5]. Specific results are easily read off Table I and Theorem 2.1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 579 - 587
Copyright
Copyright © Canadian Mathematical Society 1971

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