Real line bundles on spheres
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- by Allan L. Edelson
- Proc. Amer. Math. Soc. 27 (1971), 579-583
- DOI: https://doi.org/10.1090/S0002-9939-1971-0301016-8
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Abstract:
In a recent paper the author proved a classification theorem for Atiyah-real vector bundles on spaces with free involutions. This result is now applied to the group of Atiyah-real line (i.e., one-dimensional) bundles on spheres, denoted ${L_R}({S^n})$. It is proved that such bundles are classified by maps into a complex quadric $Q{C^n}$. Using this classification it is proved that ${L_R}({S^1}) = 0$ and that for $n \geqq 3$ the groups are all isomorphic.References
- M. F. Atiyah, $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386. MR 206940, DOI 10.1093/qmath/17.1.367
- Allan L. Edelson, Real vector bundles and spaces with free involutions, Trans. Amer. Math. Soc. 157 (1971), 179–188. MR 275417, DOI 10.1090/S0002-9947-1971-0275417-5
- Peter S. Landweber, Fixed point free conjugations on complex manifolds, Ann. of Math. (2) 86 (1967), 491–502. MR 220317, DOI 10.2307/1970612
- Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 579-583
- MSC: Primary 55F15; Secondary 55F50
- DOI: https://doi.org/10.1090/S0002-9939-1971-0301016-8
- MathSciNet review: 0301016