I am wondering if there is any computation of stable homotopy groups of $\mathbb{R}P^{\infty}\wedge \mathbb{R}P^{\infty}$ in low dimensions? I would be very grateful for any reference.

2$\begingroup$ Does $\mathbb{R} P$ equal $\mathbb{R} P^\infty$? $\endgroup$– Mark GrantOct 31, 2019 at 11:26

$\begingroup$ @Mark Grant yes $\endgroup$– user51223Oct 31, 2019 at 16:35

$\begingroup$ Upto how many dimensions? $\endgroup$– PrasitOct 31, 2019 at 23:01

$\begingroup$ @Prasit Something reasonable. Say, up to the same dimension that we know stable stems, at least at the prime $2$. $\endgroup$– user51223Nov 1, 2019 at 7:33
2 Answers
Bob Bruner and Christian Nassau both have code that can compute the Adams charts efficiently. For example, Bob has a chart for $\mathbb{R}P^2\wedge\mathbb{R}P^2$ at http://www.rrb.wayne.edu/cohom/index.html, and he might be able to do $\mathbb{R}P^\infty\wedge\mathbb{R}P^\infty$ with little effort if you asked him nicely. But I don't know where you would find information about the Adams differentials.

$\begingroup$ Thanks for this. I did not know of the charts. Meanwhile, I used some AHWspectral sequence to do some computations in very low dimensions. From this computations, I learn that the differentials in this spectral sequence also could be very difficult to recongise/identify. $\endgroup$ Nov 6, 2019 at 12:33
I just wanted to chime in to plug my own Ext calculator (a joint venture with Dexter Chua). You can see an interactive calculation of $\mathbb{R}\mathrm{P}^4\wedge\mathbb{R}\mathrm{P}^4$ here:
There's a list of other example modules that you can resolve here: https://hoodmane.github.io/rust_ext/
You can interactively add Adams differentials and it will propagate them using a few basic rules, but you'll mostly need to work them out yourself.