Definition talk:Freely Homotopic Loops

Definition collision
I should point out that the definition of freely homotopic given by Lee is different from the definition Definition:Free Homotopy, which does not include the requirement that


 * $\forall t \in \closedint 0 1 : \map H {0, t} = \map H {1, t}$

The extra requirement of $H$ comes from the idea that for all $t \in \closedint 0 1$, the map $s \mapsto \map H {s,t}$ should itself be a loop. Of course, this only works when $H$ is a homotopy between two loops. This is why the general definition does not include this requirement.

We should find a way to separate these two definitions - maybe make a subpage to Definition:Free Homotopy, where we explain the difference between freely homotopic and freely homotopic loops. Any suggestions? --Anghel (talk) 23:53, 28 November 2022 (UTC)


 * Fiddly. The short answer is that we need to establish a source for Definition:Free Homotopy. --prime mover (talk) 06:26, 29 November 2022 (UTC)


 * Good point. There are plenty of sources that define homotopy - I could cite Munkres' Topology. However, Lee's book is the first citation of a hardcopy source for the definition of free homotopy that I have seen.


 * How about we change the existing Definition:Free Homotopy so it only defines homotopy and homotopic; we cite Munkres, and then we add a 'Also Known As' section where we explain that some sources use the name of free homotopy? --Anghel (talk) 19:56, 29 November 2022 (UTC)


 * I think homotopy is too vague. You should distinguish free homotopy and based homotopy. Under free homotopy, there is at least a difference between Definition:Free Homotopy/Continuous mapping and Definition:Free Homotopy/Loop. However, I am not very sure because this topic is really large. --Usagiop (talk) 00:01, 30 November 2022 (UTC)


 * There are some sources that only define homotopy and do not use the phrase free homotopy, such as Munkres. What all definitions of a homotopy $H$ have in common, whether the homotopy is free, based, nul- or path, is that they all describe a continuous mapping $H: X \times \closedint 0 1 \to Y$ between two continuous functions $f,g:X \to Y$ such that


 * $H \left({x, 0}\right) = f \left({x}\right)$
 * $H \left({x, 1}\right) = g \left({x}\right)$


 * By the way, what you refer to as based homotopy is the same as (or a special case of) stationary homotopy that User:Julius refers to, which is the same as Definition:Relative Homotopy. --Anghel (talk) 00:07, 1 December 2022 (UTC)


 * OK, so, this is only the special case $X=\mathbb S^1$. --Usagiop (talk) 09:12, 1 December 2022 (UTC)

Which book by Lee do you have in mind? The one about topological manifolds contains an entire chapter about homotopies where it defines two maps being freely homotopic when the homotopy is not stationary. The stationary homotopy is defined to be such that for $X, Y$ being topological spaces and $A\subseteq X$ together with maps $f,g : X \to Y$ the homotopy is stationary on $A$ if $\forall x \in A : \forall t \in \closedint 0 1 : \map H {x, t} = \map f x$.--Julius (talk) 00:59, 30 November 2022 (UTC)


 * I'm presuming the one cited on this page. --prime mover (talk) 06:19, 30 November 2022 (UTC)


 * Exactly (and I am only aware of this book through the citation here). --Anghel (talk) 00:07, 1 December 2022 (UTC)

Could we make a shopping list for the homotopy theory? I could deviate from my path to anchor most pressing pages on homotopy. For existent pages, if they match the book, then I simply add a link to the book. If there is some difference, I could create new pages and then invite to rename/transclue/absorb them.--Julius (talk) 23:06, 30 November 2022 (UTC)