A cohomology theory based upon self-conjugacies of complex vector bundles

Abstract

Listen

Translate article

Let £: E—>X be a complex w-plane bundle. A self-conjugacy of £ is a map % • E-+E which is a norm-preserving bundle equivalence on the underlying 2w-dimensional real bundle and has the property that x(Xe) = Xx(^) for e(E.E and X complex. If Xi is a self-conjugacy of £*, define (£1, xi) to be equivalent to (£2, X2) if there is a bundle equivalence /x: £1—>E2 with M^^M homotopic to xi; i-e. there is a continuous map H: EiXi—>£i with if| EiXO — M^^M» u | .E iXl=Xi> a n d i J | £ i X / is a self-conjugacy of £1 for each / £ / . If X is a self-conjugacy of £, x niay be interpreted as a cross-section of the bundle £' associated to £, whose fibre is the space of 1-1 normpreserving self-conjugacies of the fibre of £. The fibre of £', denoted V(n), is homeomorphic to U{n)1 the unitary group, but there is no canonical homeomorphism. A noncanonical homeomorphism is given by left or right composition with any element of V(n). Let W(n) be the total space of the V(n) bundle associated to the universal bundle over BU(n). Then if/: X—>BU(n) is a classifying map for £ and x is a self-conjugacy of £, x interpreted as a cross-section of £' gives a lifting of ƒ to a map g: X—>W(n). Conversely any such lifting of ƒ gives a self-conjugacy of £.