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Let $n>1$. The homology of the free loop space $\Lambda S^n$ of the sphere $S^n$ contains two torsion if $n$ is even. Thus the fibration $$ \Omega S^n\rightarrow \Lambda S^n\rightarrow S^n $$ is not trivial if $n$ is even (Here $\Omega S^n$ denotes the based loop space.).

The odd dimensional spheres do not have torsion in the homology of the free loop space and one can compute that $H_*(\Lambda S^{2k+1};\mathbb Z)\cong H_*(\Omega S^{2k+1}\times S^{2k+1};\mathbb Z)$. Hence the homology does not obstruct the existence of a trivialization of the free loop fibration. Indeed $\Lambda S^3$ and $\Omega S^3\times S^3$ are homeomorphic, which can be proven using the group structure on $S^3$.

Is the free loop space fibration always trivial for odd dimensional spheres ? My guess would be that this is not the case, with a possible exception of $S^7$.

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    $\begingroup$ It definitely splits for $S^7$ since it's an H-space: mathoverflow.net/a/207856/36146 $\endgroup$ – Najib Idrissi May 31 at 12:38
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    $\begingroup$ Very nice question! One observation is that you can't use cup products in cohomology to detect non-splitting, by a result of Menichi. Also it seems that $LS^n$ and $\Omega S^n\times S^n$ are rationally homotopy equivalent for $n$ odd (by looking at the minimal models). This doesn't quite do it though. $\endgroup$ – Mark Grant May 31 at 15:19
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    $\begingroup$ I was wondering if the fact that the Whitehead square $\[\iota_n,\iota_n]\in \pi_{2n-1}(S^n)$ is non-trivial if $n\neq 1,3,7$ can be used to detect non-splitting. It certainly gives non-splitting of the analogous fibration where maps from $S^1$ are replaced by maps from $S^p$. $\endgroup$ – Mark Grant May 31 at 15:23
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    $\begingroup$ Your desired equivalence exists after p-localizing for p odd, because odd-dimensional spheres are p-local H-spaces. $\endgroup$ – skd May 31 at 16:37
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    $\begingroup$ Here is the link to the paper that was already mentioned: jstor.org/stable/43741886?seq=1#metadata_info_tab_contents $\endgroup$ – Jens Reinhold May 31 at 19:25
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I am grateful to Tobias Barthel, who sent me the following paper of J. Aguadé:

"On the space of free loops of an odd sphere". Pub. Mat. UAB No 25, June 1981.

Aguadé proves the following theorem

Theorem: The following are equivalent:

  1. $\Lambda S^{2n+1}\simeq S^{2n+1}\times \Omega S^{2n+1}$;
  2. The free loop space fibration is homotopically trivial
  3. $n=0,1,3$.

That 3 implies 2 is due to the fact that these spheres are $H$-spaces, which was already noted. 2 implies 1 is trivial. Hence the only thing Aguadé shows is that 1 implies 3.

For this it is assumed that there is a map $f$ inducing a homotopy equivalence. From this there is an induced map $h:S^1\times S^{2n+1}\times \Omega S^{2n+1}\rightarrow S^{2n+1}$. Aguadé applies the Hopf construction to this map to get a map $$ \tilde g:S^{2n+1}*(S^1\times S^{2n})\rightarrow S^{2n+2}. $$ The space on the left is a wedge of spheres $S^{2n+3}\vee S^{4n+2}\vee S^{4n+3}$, thus there is an induced map $g:S^{4n+3}\rightarrow S^{2n+2}$. Aguadé shows that this map has Hopf invariant one, hence the result follows.

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