Timeline for The free loop space of spheres

9 events

when toggle format	what		by	license	comment
Jun 2, 2019 at 13:56	vote	accept	Thomas Rot
Jun 2, 2019 at 13:55	answer	added	Thomas Rot		timeline score: 17
May 31, 2019 at 19:25	comment	added	Jens Reinhold		Here is the link to the paper that was already mentioned: jstor.org/stable/43741886?seq=1#metadata_info_tab_contents
May 31, 2019 at 16:37	comment	added	skd		Your desired equivalence exists after p-localizing for p odd, because odd-dimensional spheres are p-local H-spaces.
May 31, 2019 at 15:23	comment	added	Mark Grant		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 $S^p$.
May 31, 2019 at 15:19	comment	added	Mark Grant		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.
May 31, 2019 at 14:46	history	edited	Thomas Rot	CC BY-SA 4.0	added 10 characters in body
May 31, 2019 at 12:38	comment	added	Najib Idrissi		It definitely splits for $S^7$ since it's an H-space: mathoverflow.net/a/207856/36146
May 31, 2019 at 11:47	history	asked	Thomas Rot	CC BY-SA 4.0