Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-06-09T02:17:56.921Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Adams, J. F., J(X)-IV, Topology 5 (1966), 21–72.Google Scholar
2. Antonelli, P. L., On the stable diffeomorphism of homotopy spheres in the stable range, n ≤ 2p, Bull. Amer. Math. Soc. 75 (1969), 343–346.Google Scholar
3. Barratt, M. G. and Mahowald, M. E., The metastable homotopy of 0(n), Bull. Amer. Math. Soc. 70 (1964), 758–760.Google Scholar
4. Bott, R., The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337.Google Scholar
5. DeSapio, R., Differentiable structures on a product of spheres, Comment. Math. Helv. 44 (1969), 61–69.Google Scholar
6. Haefliger, A., Plongements differentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47–82.Google Scholar
7. Hilton, P. J., A note on the P-homomorphism in homotopy groups of spheres, Proc. Cambridge Philos. Soc. 59 (1955), 230–233.Google Scholar
8. Hoo, C. S., Homotopy groups of Stiefel manifolds, Ph.D. thesis, Syracuse University, Syracuse, New York, 1964.Google Scholar
9. Hoo, C. S. and Mahowald, M., Some homotopy groups of Stiefel manifolds, Bull. Amer. Math. Soc. 71 (1965), 661–667.Google Scholar
10. Hsiang, W. C., Levine, J., and Szczarba, R., On the normal bundle of a homotopy sphere embedded in Euclidean space, Topology 13 (1965), 173–181.Google Scholar
11. Kervaire, M., An interpretation ofG. Whitehead's generalization of Hopfs invariant, Ann. of Math. (2) 69 (1959), 345–365.Google Scholar
12. Kervaire, M., Higher dimensional knots, Differential and combinatorial topology, A symposium in honor of M. Morse (Princeton Univ. Press, Princeton, N.J., 1962).Google Scholar
13. Kervaire, M., Some non-stable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161–169.Google Scholar
14. Kervaire, M. and Milnor, J., Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537.Google Scholar
15. Kosinski, A., On the inertia group of ir-manifolds, Amer. J. Math. 89 (1967), 227–248.Google Scholar
16. Mahowald, M., Metastable homotopy of Sn, Mem. Amer. Math. Soc, No. 72, (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
17. Mahowald, M., Some Whitehead products in Sn, Topology 4 (1965), 17–26.Google Scholar
18. Massey, W., On the normal bundle of a sphere imbedded in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), 959–964.Google Scholar
19. Mimura, M., On the generalized Hopf homomorphism and higher composition. Part II. Πn+1(Sn)for i = 21 and 22, J. Math. Kyoto Univ. (1935-65), 301–326.Google Scholar
20. Mimura, M. and Toda, H., The (n + 20)th homotopy groups of n-spheres, J. Math. Kyoto Univ. 3 (1963), 37–58.Google Scholar
21. Smale, S., On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399.Google Scholar
22. Toda, H., Composition methods in homotopy groups of spheres, Annals of Math. Studies No. 49 (Princeton Univ. Press, Princeton, N.J., 1963).Google Scholar