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I understand the general notion of twist like that of a mobius band(like a cylinder cut open and the glue the edges together with a "twist") , and the torsion in the context of algebraic structures like groups, in which a modulo-n torsion indicate when we multiply some element with n, we get the identity(please correct me if there is an error in my understanding).

So in the context of topological groups, can these ideas be related, like in the case of a lie group, where we can talk about both twist and torsion and whether these ideas turn out to be related?

Eden Zane
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    You cannot talk about "twist" without defining it; to relate it to torsion in a group (or, maybe, to torsion of a connection...), or to do anything with it, you need to begin with formulating that vague idea in a precise mathematical manner. – xsnl Nov 07 '23 at 06:23
  • I am sorry, please see the edit. I mean the general notion of a twist is quite clear without defnining it. I don't mean twist as a mathematical object. – Eden Zane Nov 07 '23 at 10:38
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    If it's quite clear, there should be a definition available. If on the other hand you can't articulate a definition, then it is not actually clear. – John Palmieri Nov 07 '23 at 17:10
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    Maybe twist refers to orientation? – Vincent Boelens Nov 07 '23 at 19:41
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    Your examples of twist are just that: examples. If you try to generalize this notion you will quickly see it is quite hard. It requires the concept of cutting, glueing, some sort of orientation and its reversing. Maybe it can be done, but either way it doesn't seem to be related to torsion at all. – freakish Nov 09 '23 at 07:50
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    The twist, as you describe it, is gluing edges with an orientation-reversing map. Algebraically, reversing the orientation is usually represented via flipping the sign. I don't see how this relates to torsion. I might be missing something and/or handwaving though – NG_ Nov 09 '23 at 07:51
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    https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Looking for meaning of torsion, it too seems to come from algebraic topology and Rotman uses the word twist in his explanation. See Jim's answer. Alas, no precise definition is given. Perhaps "twist" is some intuitive explanation for the concept for torsion/torsion-free. – AlvinL Nov 09 '23 at 07:52

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As others have explained in the comments, you haven't quite defined what a "twist" is. However, perhaps what follow will still be helpful and matches what your intuitive picture of "twist" means.

Given a (path connected, semilocally simply connected) space $X$, you can construct its orientation cover $p : \tilde{X} \to X$, which is a double cover of $X$. If $X$ is orientable, then $\tilde{X} = X \times {\pm 1}$ is a trivial double cover (and disconnected). There's no "twist" and no "torsion" here.

If $X$ is not orientable, then the cover is nontrivial. The group of deck transformations of $p$ is a subgroup $\Gamma$ of index $2$ of $\pi_1(X)$. The quotient is $\mathbb{Z}/2\mathbb{Z}$ and contains a torsion element of order $2$. You can think of this element as measuring "twists": if you have a loop $\gamma$ that lifts to a path which isn't a loop (i.e., it's a nonzero element in $\pi_1(X)/\Gamma$) then there is some kind of "twist" along this loop. But if you run twice along the loop, you'll always get zero, so the "twist" is always going to be "2-torsion" (in the group-theoretic sense).

tl;dr: When you say "twist", you're probably thinking of non-orientability, and this corresponds to 2-torsion in a certain quotient of the fundamental group.

As far as I can tell, it's unrelated to (2-)torsion in a topological group. Lie groups are always orientable.

Najib Idrissi
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