## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 5, Number 1 (2003), 161-212.

### 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 $KO_G$) 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 5, Number 1 (2003), 161-212.

**Dates**

First available in Project Euclid: 13 February 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1139839931

**Mathematical Reviews number (MathSciNet)**

MR1989617

**Zentralblatt MATH identifier**

1032.55016

**Subjects**

Primary: 55Q91: Equivariant homotopy groups [See also 19L47]

Secondary: 19L20: $J$-homomorphism, Adams operations [See also 55Q50] 55Q50: $J$-morphism [See also 19L20] 55R91: Equivariant fiber spaces and bundles [See also 19L47]

#### Citation

French, Christopher. The equivariant $J$-homomorphism. Homology Homotopy Appl. 5 (2003), no. 1, 161--212. https://projecteuclid.org/euclid.hha/1139839931