# 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

## 2 Answers

The terminology in topology seems to go back a long way: Über die Torsionszahlen von Produktmannigfaltigkeiten (Mathematische Annalen 1924) is the paper in which Künneth sets out to show how the torsion in homology behaves in a product, and there he footnotes earlier work of Poincaré and Tietze (his advisor) on the "Torsionszahlen". So it is a fair bet the terminology goes back at least to Tietze. In terms of the question "algebra or topology first?", I'd therefore lay bets on topology, bearing in mind that homology wasn't clearly formulated as taking values in abelian groups until a little later. The more general idea of "twisting" probably awaited the theory of bundles, a little later again.

There is some quite detailed discussion of the history and Emmy Noether's intervention in http://math.unice.fr/~nbasbois/Articles/groupe_homologie2.pdf. A very interesting point, for algebraists, is that the issue is tied up with the structure theory of finitely-generated abelian groups and the chicken-and-egg question of whether you derive it from the theory of elementary divisors of integral matrices, or vice versa (which would be the modern approach in abstract algebra, despite what is said on "Is there a slick proof of the classification of finitely generated abelian groups?"). Psychologically this makes quite a difference to what one is looking for as a canonical form. Poincaré definitely approached it from elementary divisors, and as the paper explains, the introduction of abelian groups as adjuncts to the incidence matrix had to be "justified" to topologists.

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See Stillwell's Classical Topology and Combinatorial Group Theory, pp. 170--171 for a discussion of the historical origins of torsion, which of course is in algebraic topology, not abstract algebra. Find the title on Google books and search for the word torsion.

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