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?


1 Answer 1


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 $\mu^*_p \circ \beta^p$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.