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I would like to know where about I can find the most updated results on the unstable groups $\pi_{2k+1}P_{k+1}$ and $\pi_{2k}P_k$. I think there would be definitely computations Mahowald's AMS memoir, but I presume there has been some work after this masterpiece!

ADDED In particular, I am interested in groups $\pi_{2k+1}P_{k+1}^{2k}$ and $\pi_{2k}P_k^{2k-1}$, if by any means computations easier. In particular, any result providing some vanishing conditions is interesting.

I also wonder how is known about the stable homotopy groups $\pi_{2k+1}^sP_{k+1}$ and $\pi_{2k}^sP_k$?!

Thank you in advance!

EDIT Here, $P$ is the infinite dimensional real projective space, $P^n$ is its $n$-skeleton and $P_n:=P/P^{n-1}$ for $n>0$ and $P_0=P_+$.

Further edit I think I have managed to show that $\pi_{2k+1}P_{k+1}^{2k}$ always has a summand of $\mathbb{Z}/2$ which is precisely the kernel of the stabilisation map $\pi_{2k+1}P_{k+1}^{2k}\to\pi_{2k+1}^sP_{k+1}^{2k}$, bearing in mind that I work at the prime $2$. By similar arguments, one can show that $\pi_{2k+1}\Sigma P_{k}^{2k-1}$ also has a summand which is isomorphic to $\mathbb{Z}/2$ being precisely the kernel of the stablisation map $\pi_{2k+1}\Sigma P_{k}^{2k-1}\to \pi_{2k}^s P_{k}^{2k-1}$.

So, whether there exists some vanishing conditions which forces $\pi_{2k+1}P_{k+1}^{2k}\simeq 0$ definitly has negative answer. It is likely, that the argument for $\pi_{2k+1}\Sigma P_k^{2k-1}$ shows that $\pi_{2k}P_k^{2k-1}$ is never nontrivial.

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    $\begingroup$ Real proyective spaces? They have the same higher homotopy groups as the corresponding spheres (their universal covers). $\endgroup$ – Fernando Muro Feb 6 '18 at 14:27
  • $\begingroup$ @FernandoMuro Are you sure that your statement is about $P_k$ or $P_{k+1}$? These are truncated spaces with $P_m=P/P^{m-1}$. I know of the isomorphism $\pi_iS^n\simeq\pi_iP^n$ for $i>1$. $\endgroup$ – user51223 Feb 6 '18 at 15:04
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    $\begingroup$ If that's what you meant, then no, that's why I was asking. $\endgroup$ – Fernando Muro Feb 6 '18 at 15:06

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