The word “torsion” and its connection to geometry and homology

In an $R$-module $M$, an element $m \in M$ is said to be torsion if $am = 0$ for some $a \in R$ with $a \neq 0$.

Also, for a non-orientable (closed) surface such as the projective plane or the Klein bottle, there is a $\mathbb Z/2$ term in the first homology. This part is said to detect the "twisting" in the surface.

So this leads to the plausible notion that the origin of the word "torsion" in algebra is related to this "torsion" or "twisting" from topology.

Unfortunately the difficulty is that this is not exactly true. For example, the Möbius band is surely a twisted object. But it deformation retracts to the circle, and therefore its homology is very normal.

I hope somebody can shed more light on this terminology.

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First, the "twistedness" of the Mobius strip $M$ is encoded in how it fibers over $S^1$ as nontrivial topological (or smooth) line bundle. Second, the exponential sequence $0 \rightarrow O_X \rightarrow O_X^{\times} \rightarrow \mathbf{Z}/2\mathbf{Z} \rightarrow 0$ for any topological space or smooth manifold $X$ (with $O_X$ the sheaf of continuous or smooth functions) yields an isomorphism ${\rm{Pic}}(X) = {\rm{H}}^1(X,O_X^{\times}) \rightarrow {\rm{H}}^1(X,\mathbf{Z}/2\mathbf{Z})$. Thus, the line bundle $M$ represents the nontrivial class in ${\rm{H}}^1(S^1,\mathbf{Z}/2\mathbf{Z})$. – BCnrd May 25 '10 at 12:34
[harmless typo: In the above I should have said "paracompact Hausdorff topological space or manifold..." so that one has partitions of unity so as to kill high cohomology of $O_X$.] – BCnrd May 25 '10 at 12:36