Real vector bundles and spaces with free involutions

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.