# K-theory of classifying spaces

Can someone help me calculate the following groups in $$K$$-theory

1) $$KU^0 (B\mathbb{S}^1)$$

2) $$KU^0 (\mathbb{RP}^\infty)$$

where $$B \mathbb{S}^1$$ is the classifying space of $$\mathbb{S}^1$$

Thank you

• What $K$ theory do you mean (as your spaces are non-compact). – Thomas Rot Jun 12 '19 at 21:46
• $BS^1=\mathbb{CP}^\infty$. You should be a bit careful to say what the K theory is of a non-compact space. In Atiyah's K-theory book corollary 2.5.4 it is proven that $K(\mathbb{CP}^n)\cong \mathbb{Z}[t]/(1-t)^n$. – Thomas Rot Jun 13 '19 at 7:27

Let $$G$$ be a compact Lie Group and let $$R_{\mathbb{C}}(G)$$ be its representation ring of finite dimensional continuous complex representations and let $$I$$ be the augmentation ideal i.e. the kernel of the canonical morphism $$R_{\mathbb{C}}(G)\to \mathbb{Z}$$ which sends every representation to its dimension. If we let $$BG$$ denote the classifying space of this group, then the content of the Atiyah-Segal Completion theorem is that $$K^0(BG)\cong R_{\mathbb{C}}(G)_{\hat{I}}$$ and that $$K^{1}(BG)=0$$, where $$R_{\mathbb{C}}(G)_{\hat{I}}$$ is the completion of the representation ring at $$I$$. We note that the Completion theorem says more than just this, but this will suffice for our purposes.
As a computational tool note that if $$R$$ is a noetherian ring and $$I=(a_1,\ldots, a_n)$$, then $$R_{\hat{I}}=R[[x_1,\ldots,x_n]]/(x_1-a_1,\ldots, x_n-a_n).$$
Now in the harder second case we have that $$RP^{\infty}=B\mathbb{Z}/2\mathbb{Z}$$. Notice that the representation ring of $$\mathbb{Z}/2\mathbb{Z}$$ can canonically be identified with its character ring, since each irreducible representation is one dimensional and there are two of them. This shows that we must have $$R_{\mathbb{C}}(\mathbb{Z}/2\mathbb{Z})=\mathbb{Z}[x]/(x^{2}-1)$$. The augmentation ideal is generated by $$(x-1)$$. Set $$x-1=u$$, and we see that we are completing the ring $$\mathbb{Z}[u]/((u+1)^{2}-1)$$ at the ideal $$u$$ but this is just isomorphic to $$\mathbb{Z}[[u]]/((u+1)^{2}-1)$$ . Thus we have $$K^{0}(RP^{\infty})=\mathbb{Z}[[u]]/((u+1)^{2}-1)$$ and $$K^1(RP^{\infty})=0.$$
In the other case we have $$G=S^{1}$$. One can show that $$R_{\mathbb{C}}(S^{1})=\mathbb{Z}[x,x^{-1}]$$. Running the above argument once again we see that the augmentation ideal is really just $$(x-1)$$ again. Changing coordinates $$x-1=c$$ we see that we are completing the ring $$\mathbb{Z}[c+1,(c+1)^{-1}]$$ at $$(c)$$. But clearly this is just $$\mathbb{Z}[[c]]$$.
Thus $$K^0(BS^{1})=\mathbb{Z}[[c]]$$ and $$K^1(BS^{1})=0.$$