Talk:Loop space

In what sense does loop space form a quasigroup? If I have two loops f and g there need not be a loop h such that fh = g. Loop space forms a nonassocitive, noncommutative magma, but surely not a quasigroup. -- Fropuff

Yes, you're right. Got my definitions mixed up. -- Staecker 01:30, 20 October 2005 (UTC)Reply

In the unbased case, the left adjoint to the loop functor is given by ordinary product with the circle (since the unbased loop functor is by definition the mapping space of maps from the circle). So IMHO the article cannot be quite correct in saying that the loop space functor is right adjoint to the unreduced suspension. What do you guys think? - Saibot2 21:36, 13 February 2007 (UTC)Reply

Saibot2 is right, the free loop functor (I think that's a more common name than unbased loop, but less descriptive) is right adjoint to the cartesian product. I also think that ΩX is a notation used only for the based loop space; the free loops are usually denoted L(X) or ${\displaystyle {\mathcal {L}}(X)}$ . But maybe the usage outside of topology differs. -- Tilmanbauer (talk) 19:25, 19 February 2008 (UTC)Reply

"The free loop space construction is right adjoint to the cartesian product with the circle" Is the smash product really cartesian? The smash product page says only that it is symmetric monoidal... 83.160.106.234 (talk) 04:37, 8 July 2013 (UTC)Reply

Not sure, but perhaps its this: it is the reduced suspension that is the smash product between circle and space. The adjoint to the reduced suspension is the based loop space. For the un-based loop space, its adjoint to the cartesian product, not the smash product. User:Linas (talk) 16:58, 19 November 2013 (UTC)Reply

An important result that needs to be added:

Relation between homotopy groups of ${\displaystyle X}$  and those of its loop space: ${\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)}$  ^[1].

More generally, ${\displaystyle [\Sigma Z,X]\approxeq [Z,\Omega X]}$ , where ${\displaystyle [A,B]}$  is the set of homotopy classes of maps ${\displaystyle A\rightarrow B}$  (with usual group structure when ${\displaystyle A,B}$  are pointed), and ${\displaystyle \Sigma A}$  is the suspension of A. ^[2] Note that setting ${\displaystyle Z=\mathbb {S} ^{k-1}}$  gives the earlier result.

- Subh83 (talk | contribs) 22:03, 31 October 2013 (UTC)Reply

Looks like you or someone already added that to the article. Note that JP May has an entire chapter(s) on the long exact sequences that arise in these constructions, and some kind of summary of that could be nice for this article. User:Linas (talk) 17:02, 19 November 2013 (UTC)Reply

ya, that's right. This can be expanded upon.

I added a clarification on the group structure of [A,B].- Subh83 (talk | contribs) 05:04, 28 December 2013 (UTC)Reply

In order to include the phrase 'applying Ω a number of times' in the article, I think we need to first describe how taking the loop space of a topological space can be seen as an operator from the class of topological spaces to itself, which we denote Ω. if this is a very weird way of putting things, then I suggest 'applying Ω a number of times' be changed to 'repeatedly taking the loop space of X'. I am by no means an expert so somebody more experienced with loop spaces should make the edit. Joel Brennan (talk) 14:19, 5 April 2019 (UTC)Reply