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Homotopy classification of connected sums of sphere bundles over spheres, I

Published online by Cambridge University Press:  22 January 2016

Hiroyasu Ishimoto
Hiroyasu Ishimoto*
Affiliation:
Department of Mathematics, Faculty of Science, Kanazawa University
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In the classification problems of manifolds, the connected sums of sphere bundles over spheres appear frequently. In fact, the manifolds with certain tangential and homotopy properties come to such connected sums (cf. Tamura [15], [16], Ishimoto [6], [8], [9]). Motivated by those, in this paper and the subsequent paper, we classify connected sums of sphere bundles over spheres up to homotopy equivalence by extending the results of I. M. James and J. H. C. Whitehead [10], [11], which correspond to the case that the connected sums of the above happen to be single sums. We also use Wall [17] in the case when the fibres and the base spaces of bundles are same dimensional.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 83 , October 1981 , pp. 15 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Adams, F., On the groups J(X)—IV , Topology, 5 (1966), 21–71.Google Scholar
[2] Barcus, W. D. and Barratt, M. G., On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc., 88 (1958), 57–74.Google Scholar
[3] Eckmann, B. and Hilton, P. J., Décomposition homologique d’un polyèdre simplement connexe, C. R. Acad. Sci. Paris, 248 (1959), 2054–2056.Google Scholar
[4] Hilton, P. J., On the homotopy groups of the union of spheres, J. London Math. Soc., 30 (1955), 154–172.Google Scholar
[5] Hirsch, M. W., Differential Topology, Springer-Verlag, 1976.Google Scholar
[6] Ishimoto, H., On the structure of (n—2)-connected 2n-dimensional π-manifolds, Publ. R.I.M.S. Kyoto Univ., 5 (1969), 65–77.Google Scholar
[7] Ishimoto, H., Representing handlebodies by plumbing and surgery, ibid., 7 (1972), 483–510.Google Scholar
[8] Ishimoto, H., On the classification of (n—2)-connected 2n-manifolds with torsion free homology groups, ibid., 9 (1973), 211–260.Google Scholar
[9] Ishimoto, H., On the classification of some (n—3)connected (2n—1) manifolds, ibid., 11 (1976), 723–747.Google Scholar
[10] James, I. M. and Whitehead, J. H. C., The homotopy theory of sphere bundles over spheres (I), Proc. London Math. Soc., (3) 4 (1954), 196–218.Google Scholar
[11] James, I. M. and Whitehead, J. H. C., The homotopy theory of sphere bundles over spheres (II), ibid., (3) 5 (1955), 148–166.Google Scholar
[12] James, I. M., On the suspension sequence, Ann. of Math., 65 (1957), 74–107.Google Scholar
[13] Kervaire, M. A., Some non-stable homotopy groups of Lie groups, Illinois J. Math., 4 (1960), 161–169.Google Scholar
[14] Milnor, J. W., On simply connected 4-manifolds, Symposium Internacional de Topologia Algebrica, Mexico (1958), 122–128.Google Scholar
[15] Tamura, I., Classification des variétés differentiates, (n— 1)-connexes, sans torsion, de dimension 2n + 1 , Séminaire H. Cartan, 1962/63, n° 16–19.Google Scholar
[16] Tamura, I., On the classification of sufficiently connected manifolds, J. Math. Soc. Japan, 20 (1968), 371–389.Google Scholar
[17] Wall, C. T. C., Classification of (n—1)-connected 2n-manifolds, Ann. of Math., 75 (1962), 163–189.Google Scholar
[18] Wall, C. T. C., Classification problems in differential topology-I, Classification of handle-bodies, Topology, 2 (1963), 253–261.Google Scholar
[19] Whitehead, G. W., A generalization of the Hopf invariant, Ann. of Math., 51 (1950),192–237.Google Scholar