Real vector bundles and spaces with free involutions
Abstract: The functor , defined in , 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, is an invariant of the homotopy type of the orbit space modulo its involution. The multiplicative group of real line bundles, real in the sense of , is shown to be classified by equivariant maps into a quadric surface in complex projective space. carries a free involution and this classification is again valid for spaces with free involutions.
-  John Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405. MR 0082103, https://doi.org/10.2307/1969983
-  -, Lectures on characteristic classes, Princeton University, 1957.
-  Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
-  M. F. Atiyah, 𝐾-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386. MR 0206940, https://doi.org/10.1093/qmath/17.1.367
-  -, -theory, Lectures, Harvard University (1964), Benjamin, New York, 1967. MR 36 #7130.
-  Peter S. Landweber, Conjugations on complex manifolds and equivariant homotopy of 𝑀𝑈, Bull. Amer. Math. Soc. 74 (1968), 271–274. MR 0222890, https://doi.org/10.1090/S0002-9904-1968-11917-2
-  Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
- J. Milnor, On manifolds homeomorphic to the -sphere, Ann. of Math. (2) 64 (1956), 399-405. MR 18, 498. MR 0082103 (18:498d)
- -, Lectures on characteristic classes, Princeton University, 1957.
- D. Husemoller, Fibre bundles, McGraw-Hill, New York, 1966. MR 37 #4821. MR 0229247 (37:4821)
- M. F. Atiyah, -theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367-386. MR 34 #6756. MR 0206940 (34:6756)
- -, -theory, Lectures, Harvard University (1964), Benjamin, New York, 1967. MR 36 #7130.
- P. Landweber, Conjugations on complex manifolds and equivariant homotopy of MU, Bull. Amer. Math. Soc. 74 (1968), 271-274. MR 0222890 (36:5940)
- N. Steenrod, Topology of fibre bundles, Princeton Math. Series, vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. MR 0039258 (12:522b)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 55.30
Retrieve articles in all journals with MSC: 55.30
Keywords: Atiyah-real vector bundle, conjugate linear isomorphism, equivariant homotopy, line bundle
Article copyright: © Copyright 1971 American Mathematical Society