Homology, Homotopy and Applications

Volume 5 (2003)

Number 1

The equivariant {$J$}-homomorphism

Pages: 161 – 212

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a8

Author

Christopher French (Department of Mathematics, University of Illinois at Urbana, Urbana, Il., U.S.A.)

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 $KO_G$) 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.

Keywords

equivariant J-homomorphism, Thom classes, classifying spaces, spherical fibrations

2010 Mathematics Subject Classification

19L20, 19L47, 55R91

Full Text (PDF format)