I am currently working my way through Atiyah's paper "On K-Theory and Reality" and can't get my head around a remark stated in the paper.

In the third section $KR$-Theory is related to regular $K$-theory using so called "coefficient theories", i.e. functors $X \mapsto KR(X \times S^{p,0})$. In Proposition 3.1 Atiyah proves periodicity isomorphisms in the cases $p=1,2,4$ using the maps $\mu_p \colon X \times S^{p,0} \times \mathbb{R}^{0,p} \to X \times S^{p,0} \times \mathbb{R}^{p,0}$, $(x,s,u) \mapsto (x,s,su)$ and Bott periodicity $\beta^p \colon KR \to KR^{p,p}$.

Afterwards he claims, that these isomorphisms $$ \gamma_p = \mu_p^* \circ \beta^p \colon KR(X \times S^{p,0}) \longrightarrow KR^{-2p}(X \times S^{p,0})$$ are given by multiplication by $c_p = \gamma_p(1) = \mu^*_p(b^p \cdot 1)$ for $1 \in KR(S^{p,0})$ and $b= [H] -1$ the Bott element, since $\mu^*_p$ and $\beta^p$ are $KR(X)$-module homomorphisms.

I am struggling with this claim: Both maps are clearly $KR(X)$-module homomorphisms but to prove this claim I think one also needs to know that

  1. the external product map $KR(X) \otimes KR(S^{p,0}) \to KR(X \times S^{p,0})$ is surjective and
  2. $KR(S^{p,0}) \cong \mathbb Z$ generated by $1 \in KR(S^{p,0})$

Especially the first point troubles me. Or am I missing something here?


As it turns out my confusing resulted from a – supposed – typo: The isomorphisms $\mu^*_p$ and $\beta^p$ are in fact $KR(X \times S^{p,0})$-linear (and not just $KR(X)$-linear), which is quite easy to see by following the definitions and obviously implies the statement about $\mu^*_p \circ \beta^p$.


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