# Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.

• This might contain the answer: arxiv.org/abs/0710.5942 – Qiaochu Yuan Feb 10 '16 at 19:28
• In addition to the calculation that Matthias Wendy points out, Mahowald's memoir "The metastable homotopy of $S^n$" does a lot of calculation with the Atiyah-Hirzebruch spectral sequence for stable homotopy groups of projective spaces (because this calculation connects to the EHP spectral sequence). – Tyler Lawson Feb 10 '16 at 19:50

The following paper contains a list of stable homotopy of projective spaces in dimensions $\leq 8$:
In particular, $\pi^s_5=0$ and $\pi^s_6=\mathbb{Z}/2$.