Torsion in geometry v.s. Torsion in topology?

Question

1. Torsion in geometry:

• There are meanings of torsion of a curve https://en.wikipedia.org/wiki/Torsion_of_a_curve

• Torsion tensor in Riemannian geometry https://en.wikipedia.org/wiki/Torsion_tensor

2. Torsion in topology:

• such as analytic torsion https://en.wikipedia.org/wiki/Analytic_torsion#:~:text=Reidemeister%20torsion%20was%20the%20first,used%20to%20classify%20lens%20spaces.

• Reidemeister Torsion https://mathworld.wolfram.com/ReidemeisterTorsion.html

• torsion in cohomology https://mathoverflow.net/questions/22583/why-torsion-is-important-in-cohomology

My question is that:

Whether torsion in geometry is related to the torsion in topology? Why they are all called Torsions?

Answer 1

The word literally means "twisting," and that's the sense in which it's used in the geometric context. The algebraic sense of torsion (i.e., elements of finite order in a group or similar construction) comes from the idea of torsion elements in fundamental groups or homology, where a torsion element is one that's "almost" flat. I've seen an early example of the usage describing torsion elements as detectors of or obstacles to nonorientability, which further suggests the "twisting" idea. (Probably, anyway. I've seen a few different explanations of where the word comes from, though they all share the same fundamental idea.) By extension, it's a name for things that are "almost" trivial. Reidemeister torsion, for example, is defined for acyclic complexes and measures (in a more technical sense) exactly how easy it is to trivialize them. It can distinguish, for example, spaces that are not actually homeomorphic but are just homotopy equivalent; compare that to the condition of an element in a group being trivial versus having some positive power of it trivial.

Answer 2

Group elements of finite order generate a cyclic group by considering powers of that element. For some group elements of order $n$ this will be isomorphic to the $n$-th roots of unity in the complex plane under multiplication. These roots, like all points on the complex unit circle, act on $\mathbb{C}$ by rotating them through the argument of the root. By taking powers of these roots you come back round to the start as the hour hand on a clock might. This ties elements of finite order intimately to the complex circle and rotations, and so torison is a natural term in this environment. A great deal of analytic geometry began in the complex plane and it often motivates nomenclature so I assume this is why.