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What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values of $r$, $s$, $n$, $m$.

There are some obvious cases such as $r=s=0$, or $r=s=1$ in which case the James splitting gives a lot of useful information. Rationalization also goes a long way towards solving the problem.

I've gathered it's a pretty difficult problem, but are there any special cases that are known? The simplest non-trivial case would probably be $[\Omega^2S^n,\Omega^2S^m]$.

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    $\begingroup$ I am not an expert on this. But, some of this might be helpful. 1- As pointed out the problem seems very difficult to me. I think some of the literature on the EHP SS computation can be interpreted to give information on this. For instance, Milgram's LNM "Unstable Homotopy from stable point of view" might provide some information. 2- On the other hand, unstable ASS is meant to compute $[X,Y]$ ... 3- I was trying $r=s=n=m$ case, and it seems there is an injection $\pi_nS^n\to [\Omega^nS^n,\Omega^nS^n]$ given by sending a homotopy class $[f]$ to $[\Omega^nf]$. $\endgroup$
    – user51223
    Jun 30, 2016 at 8:02
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    $\begingroup$ If you allow stablisation, when $s\geqslant r$ then one might be able to use Snaith splitting for $\Sigma^r\Omega^rS^n$ which may give some insight. On the other hand, at the prime $p=2$, one may use James fibration by applying $[\Omega^rS^n;-]$ to fibre sequences $\Omega^{s+2}S^{2m+1}\to\Omega^sS^m\to\Omega^{s+1}S^{m+1}\to\Omega^{s+1}S^{2m+1}$. But, this is probably implicit in the EHP SS techniques for which Milgram's is a reference again. $\endgroup$
    – user51223
    Jun 30, 2016 at 8:09
  • $\begingroup$ You may also consider the stablisation and ask the same question which then turns to be a question about stable cohomotopy of $\Omega^rS^n$ and I don't know how much the latter is known... $\endgroup$
    – user51223
    Jun 30, 2016 at 8:27
  • $\begingroup$ @user51223 I don't think that there is a Snaith splitting of $\Sigma^r\Omega^rS^n$ unless $r=1$. For $r>1$, you have to go all the way to $\Sigma^\infty\Omega^rS^n$ to get a splitting. $\endgroup$ Jul 1, 2016 at 10:15
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    $\begingroup$ @NeilStrickland I thought writting `if you allow stabilisation' in that comment allows me to drop $\Sigma^\infty$ from the notation. I should have been more precise. What I really meant, as you point out, is that by Snaith splitting, when $r<n$, we have $$\Sigma^\infty\Omega^rS^n\simeq\bigvee_{k=1}^{+\infty} \Sigma^\infty D_kS^{n-r}.$$ Here, for $r$-fold loop spaces, $D_kX=F(\mathbb{R}^{r+1},k)\ltimes_{\Sigma_k}X^{\wedge k}$ is the $k$-adic construction on $X$ for a pointed space $X$. Consequently, $\Sigma^\infty\Sigma^s\Omega^rS^n$ also splits. The assumption $s\geqslant r$ is irrelavent. $\endgroup$
    – user51223
    Jul 1, 2016 at 12:39

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