The equivariant {$J$}-homomorphism

Abstract

May’s J-theory diagram is generalized to an equivariant setting. To do this, equivariant orientation theory for equivariant periodic ring spectra (such as KOG) is developed, and classifying spaces are constructed for this theory, thus extending the work of Waner. Moreover, Spin bundles of dimension divisible by 8 are shown to have canonical KOG-orientations, thus generalizing work of Atiyah, Bott, and Shapiro. Fiberwise completions for equivariant spherical fibrations are constructed, also on the level of classifying spaces. When G is an odd order p-group, this allows for a classifying space formulation of the equivariant Adams conjecture. It is also shown that the classifying space for stable fibrations with fibers being sphere representations completed at p is a delooping of the 1-component of QG(S 0 )ˆ p. The “Adams-May square,” relating generalized characteristic classes and Adams operations, is constructed and shown to be a pull-back after completing at p and restricting to G-connected covers. As a corollary, the canonical map from the p-completion of J k G to the G-connected cover of QG(S 0 )ˆ p is shown to split after restricting to G-connected covers.