\title{Equivariant forms of connective K theory}
\author{J.P.C.Greenlees}
\address{School of Mathematics and Statistics, Hicks Building,
Sheffield S3 7RH. UK.}
\email{j.greenlees@sheffield.ac.uk}
Let $G$ be a group of order $p$, and
consider the following properties of a $G$-spectrum $E$.
1. $E$ is a split ring spectrum, and nonequivariantly $ku$.
2. $E[v^{-1}]\simeq K$ (equivariantly) where $v$ is the degree 2 Bott
element arising from the split structure.
3. $E$ is complex orientable (equivariantly).
4. $E^G_*$ is concentrated in even degrees.
5. $E_G^*$ is a Noetherian ring.
Property 1 simply states that $E$ is an equivariant
form of connective K-theory, and is therefore not negotiable.
\begin{thm}
There is a $G$-spectrum $ku$ with the five properties above.
Its coefficient ring is the Rees ring
$$ku_G^*=R(G)[v,y]/(vy = \chi (\alpha ), y\rho )$$
where $\alpha$ is the natural representation of $G$, $\chi (\alpha )=
1-\alpha$ is its K-theory Euler class and $\rho =1 + \alpha + \cdots
+ \alpha^{p-1}$ is the regular representation.
The Bott element $v$ is in degree 2, and the element $y$ is in degree $-2$.
\end{thm}