# $KO_*$ groups of $\mathbb{R}P^\infty$, “Snaiths” theorem for $KO$

I wonder whether anyone has taken the time to compute $$KO_*(\mathbb{R}P^\infty)$$?

The standard tools to compute these Groups in the complex case rest on the requirement for the cohomology theories $$E$$ to be complex orientable. Naturally, I looked up whether something like real orientable cohomology theories exist in the literature but found out that $$KO$$ is not real-oriented. Anyway, there is a way to "circumvent" Snaiths theorem for the spectrum $$K$$ if one is only interestd in the algebra of cooperations, in the sense that one can show that $$K_*(\mathbb{C}P^\infty) \xrightarrow{i_*} K_*K$$ is an injection of rings, where $$i$$ is induced from the inclusion $$\mathbb{C}P^\infty \simeq BU(1) \hookrightarrow BU$$. In fact, one only needs to invert the Bott element $$\beta$$ to turn it into an isomorphism, so it is a localization. This can be concluded from

Robert M. Switzer. Algebraic topology—homotopy and homology. Classics in Mathematics. Springer-Verlag, Berlin, 2002. Reprint of the 1975 original [Springer, New York; MR0385836 (52 #6695)].

17.33, which states

$$K_*K$$ is generated over $$\mathbb{Z}[u,u^{-1},v^{-1}]$$ by the polynomials $$\{p_1,p_2,\ldots\}$$.

By a process reminiscent of

J. F. Adams. Stable homotopy and generalised homology. University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics.

p. 44 we can describe the relations of the generators of $$\beta_i$$ of $$K_*(\mathbb{C}P^\infty) = K_* \{\beta_0 , \beta_1 , \ldots \}$$ such that

$$\beta_1\beta_n = n \beta_n +(n+1)\beta_{n+1}$$

and by setting

$$\binom{x}{i} = \frac{x(x-1)\cdots (x-(i-1))}{i!} \in \mathbb{Q}[x]$$

with $$x:=\beta_1$$ one can see that

$$K_*(\mathbb{C}P^\infty)\otimes \mathbb{Q}$$ is the polynomial algebra $$K_* \otimes \mathbb{Q}[x]$$ over $$K_*\otimes \mathbb{Q} = \mathbb{Q} [t,t^{-1}]$$ and $$K_*(\mathbb{C}P^\infty)$$ can be identified with the subalgebra of $$K_* \otimes \mathbb{Q}[x]$$ generated by $$\binom{x}{i}$$ for $$i=0,1,2, \ldots$$,

where we set $$\binom{x}{0}=1$$.

While snaiths theorem works on the spectrum level and the aforementioned result follows, I wonder whether a similar result holds in the real case, i.e. $$KO_*(\mathbb{R}P^\infty)[\alpha^{-1}] \cong KO_*KO$$ for some element $$\alpha \in KO_*(\mathbb{R}P^\infty)$$?