# The Equivariant J-homomorphism

## Christopher French

May's $J$-theory diagram is generalized to an equivariant setting. To do this, equivariant orientation theory for equivariant periodic ring spectra (such as $KO_G$) 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 $KO_G$-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 $Q_G(S^0)\hat{_p}$. 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 $J_G^k$ to the $G$-connected cover of $Q_G(S^0)\hat{_p}$ is shown to split after restricting to $G$-connected covers.

Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 161-212

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