Abstract Algebra – Origin of the Term ‘Torsion’

Question

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the word). So how did it acquire this usage in algebra?

I'm interested to understand the intuition behind why the word "torsion" was chosen for this notion, as well as when it was first used.

Answer 1

John Stillwell wrote that "the word 'torsion'entered the theory of abelian groups as a result of the derivation of the one-dimensional torsion coefficients by abelianization of the fundamental group in Tietze 1908" [Classical Topology and Combinatorial Group Theory, 1993, Sec. 5.1.1, p. 170]. Below is an excerpt providing further context.

The appropriate notions of "sum" and "boundary,"and the correct choice of k-dimensional manifolds admissible as basis elements, were found only after considerable trial and error. "Appropriate" initially meant satisfying the relation $B_k = B_{m-k}$ since this was the relation Poincare tried to prove in his 1895 paper. Heegaard 1898 showed this work to be in error by constructing a counterexample. Poincare then changed the definition and proved the theorem again in Poincare 1899, inventing the tool of simplicial decomposition for the purpose. He also made a thorough analysis of his error, uncovering the important concept of torsion in Poincare 1900, and exposing the breakdown of his earlier proof as failure to observe torsion.

Torsion is present when an element a does not form a boundary taken once, but does when taken more than once. An example is the curve $a$ in the projective plane $P$ which generates $\pi_1(P).$ Then $a^2$ is the boundary of a disc, though a itself does not separate $P.$ Poincare justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only in an $m$-manifold which is nonorientable, and hence twisted onto itself in some sense.

In his first topology paper, Poincare 1892 showed that the Betti numbers alone did not determine a manifold up to homeomorphism. By 1900 he was hoping that torsion numbers would supply the missing information, and his paper of that year contains a decomposition of the homology in- formation in each dimension $k$ into the Betti number $B_k$ and a finite set of numbers called $k$-dimensional torsion coefficients. Since Noether 1926 it has been customary to encode this information in an abelian group $H_k$ called the $k$-dimensional homology group, and Poincare's construction can in fact be seen as the decomposition of a finitely generated abelian group into cyclic factors (see the structure theorem 5.2). The word "torsion," which appears so inexplicably in most algebra texts, entered the theory of abelian groups as a result of the derivation of the one-dimensional torsion coefficients by abelianization of the fundamental group in Tietze 1908 (see 5.1.3. and 5.3).