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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 $\alpha \in KO_*(\mathbb{R}P^\infty)$?

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