On the homotopy type of Siefel manifolds

James, I. M.

doi:10.1090/S0002-9939-1971-0275427-3

On the homotopy type of Siefel manifolds
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Proc. Amer. Math. Soc. 29 (1971), 151-158 Request permission

Abstract:

Suppose that we have a fibre bundle where the total space has the same homotopy type as the product of the fibre and the base. When can we conclude that the bundle is trivial, in the sense of fibre bundle theory? This question arises in the classification theory of Hopf homogeneous spaces, especially in relation to Stiefel manifolds. Results are proved, using cohomology operations, which answer the question in some cases.

References

J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI 10.2307/1970147
J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
José Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720–726. MR 50278, DOI 10.1073/pnas.38.8.720
M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291–310. MR 131880, DOI 10.1112/plms/s3-11.1.291
A. Borel and J.-P. Serre, Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409–448 (French). MR 58213, DOI 10.2307/2372495
Maurice Gilmore, Some Whitehead products on odd spheres, Proc. Amer. Math. Soc. 20 (1969), 375–377. MR 234460, DOI 10.1090/S0002-9939-1969-0234460-9
P. J. Hilton and J. H. C. Whitehead, Note on the Whitehead product, Ann. of Math. (2) 58 (1953), 429–442. MR 60231, DOI 10.2307/1969746
Wu-yi Hsiang and J. C. Su, On the classification of transitive effective actions on Stiefel manifolds, Trans. Amer. Math. Soc. 130 (1968), 322–336. MR 221529, DOI 10.1090/S0002-9947-1968-0221529-1
I. M. James, On the iterated suspension, Quart. J. Math. Oxford Ser. (2) 5 (1954), 1–10. MR 61836, DOI 10.1093/qmath/5.1.1
I. M. James, Spaces associated with Stiefel manifolds, Proc. London Math. Soc. (3) 9 (1959), 115–140. MR 102810, DOI 10.1112/plms/s3-9.1.115
I. M. James and J. H. C. Whitehead, Note on fibre spaces, Proc. London Math. Soc. (3) 4 (1954), 129–137. MR 61837, DOI 10.1112/plms/s3-4.1.129
I. M. James and J. H. C. Whitehead, The homotopy theory of sphere bundles over spheres. I, Proc. London Math. Soc. (3) 4 (1954), 196–218. MR 61838, DOI 10.1112/plms/s3-4.1.196
René Thom, Espaces fibrés en sphères et carrés de Steenrod, Ann. Sci. Ecole Norm. Sup. (3) 69 (1952), 109–182 (French). MR 0054960
Emery Thomas, On functional cup-products and the transgression operator, Arch. Math. (Basel) 12 (1961), 435–444. MR 149485, DOI 10.1007/BF01650589