I posted this question some days ago [at math.stackexchange](https://math.stackexchange.com/questions/3522030/ko-groups-of-mathbbrp-infty-snaiths-theorem-for-ko), but didn't receive an answer.


I have two questions:


* 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](https://web.ma.utexas.edu/users/pedrotti.riccardo/Riccardo_Pedrotti-Generalized_orientation_and_characteristic_classes_copy.pdf). Naturally, I looked up whether something like real orientable cohomology theories exist in the literature but found out that $KO$ is [not real-oriented](https://amathew.wordpress.com/2012/05/28/the-formal-group-law-of-unoriented-cobordism/). 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 $\alpha \in KO\_\*(\mathbb{R}P^\infty)$?



$KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$
There are many ways to do this. One elementary approach is to use the Adams spectral sequence 


$Ext\_A(H^\*ko \wedge RP^\infty, F\_2) \cong Ext\_{A(1)}(H^\*RP^\infty,F\_2) \Rightarrow ko\_\*(RP^\infty) $ 


and invert the Bott map. An $A(1)$ resolution giving the answer (since $E\_2 = E\_\infty$) can be seen at the bottom of the page 


<http://www.rrb.wayne.edu/art/index.html>


We see the groups, starting with $KO\_0$, are $0, Z/2, Z/2, Z/2^\infty, 0, 0, 0, Z/2^\infty$ and repeat by Bott periodicity, with the evident action of the coefficients $KO\_\*$. (This action follows from the homological algebra of the $Ext$ calculation.)


