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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Real vector bundles and spaces with free involutions


Author: Allan L. Edelson
Journal: Trans. Amer. Math. Soc. 157 (1971), 179-188
MSC: Primary 55.30
MathSciNet review: 0275417
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Abstract: The functor $ KR(X)$, defined in [4], is a contravariant functor defined in the category of spaces with involutions. It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation. For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold. This fixed point set can be identified with the real Grassmann manifold. It is then shown that, for free involutions, $ KR(X)$ is an invariant of the homotopy type of the orbit space $ X$ modulo its involution. The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface $ Q$ in complex projective space. $ Q$ carries a free involution and this classification is again valid for spaces with free involutions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0275417-5
PII: S 0002-9947(1971)0275417-5
Keywords: Atiyah-real vector bundle, conjugate linear isomorphism, equivariant homotopy, line bundle
Article copyright: © Copyright 1971 American Mathematical Society