# The structure map of topological K-theory

This may be a silly question but I don't know the answer.

I know the construction of (equivariant) K-spectrum $KU_G$ and the periodicity of (equivariant) K-theory. But I don't know its structure maps and how they are constructed.

Can anybody show me the answer or some reference on this? Thanks very much.

## 1 Answer

That is a very reasonable question. By a $G$-spectrum $KU_G$, $G$ a compact Lie group, one should mean a genuine $G$_spectrum, so suitably indexed on representations of $G$. One gets the structure maps by use of equivariant Bott periodicity. A good sketch of how this goes, without full details, is given on pages 146-148 of Equivariant Homotopy and Cohomology Theory (available here: http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf). The relevant chapter was written by John Greenlees.