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I was reviewing my class notes and found the following:

"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band"

In our notes we used the following definition for torsion element and torsion module: An element m of an R-module M is called a torsion element if $rm=0$ for some $r\in R$. A torsion module is a module which consists solely of torsion elements

What is the relationship between torsion modules and twisted spaces? Was the definition of torsion module somehow motivated from topological considerations of twisted spaces?

I don't really see any obvious connection. I'm taking my first topology class this semester, so I apologize if this is something you learn about later in courses like algebraic topology, but I haven't been able to find any explanation of this.

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I cannot tell what you mean by «torsion modules». Do you mean $\mathrm{Tor}$ groups? – Mariano Suárez-Alvarez Sep 18 '10 at 19:12
A torsion module is a module all of whose elements are torsion. – Qiaochu Yuan Sep 18 '10 at 19:16
I edited the post to reflect the definitions we used in class. – WWright Sep 18 '10 at 19:17
up vote 7 down vote accepted

When you compute the homology groups of "twisted" spaces (which are abelian groups), you (sometimes) find that they contain non-zero torsion elements; furthermore, the presence of these particular elements in the homology is due to the twisting (in that, when you compute the homology groups, you see that is the twisting in the space that causes the calculation to give rise to torsion elements).

Since you haven't studied algebraic topology yet, I won't say more here. Hopefully, while necessarily vague, the above description gives you some feeling for the meaning of the remark in your notes.

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The definition of torsion in modules is a generalization of the definition of torsion in $\mathbb{Z}$-modules, e.g. abelian groups. Torsion in abelian groups refers to elements of finite order, and this in turn relates to topology because to any topological space we can associate abelian groups called (integral) homology groups, and torsion in these groups is suggestive of a kind of "twistedness" in the space. The simplest example of this is in the first integral homology of closed surfaces; the group has torsion if and only if the surface is non-orientable, such as the Klein bottle.

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