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The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that $2\gamma$ are boundaries. Then there's torsion in general abelian groups, rings, and modules. There's torsion in differential geometry, and analytic torsion. Lastly, there's $\mathrm{Tor}$, the left derived functor of the tensor product which is defined at least in the case of modules.

The lower dimensional $\mathrm{Tor}$ functors tell us about torsion. I don't understand what the higher ones do, but this bridge does exist. So the tensor product over of modules does poop out torsion from high above.

In differential geometry, the torsion form is often identified with a section of $TM\otimes \Lambda ^2T^\ast M$, called the torsion tensor. So formally, the tensor product pops up here too. Unfortunately

The definition of analytic torsion is beyond me entirely.

To what extent can these concepts be unified, seen as special cases of each other, or obtained from abstract nonsense?

Arrow
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  • I doubt that torsion from algebra is related to torsion in differential geometry, or to the analytic/Reidemeister torsion. It's just a name, and the idea that something has been twisted/wrung compared to something flat. – Najib Idrissi Nov 10 '15 at 08:16
  • @NajibIdrissi it's exactly because it's the same idea that I hope someone has pursued a general-enough picture which subsumes all of these special cases :) – Arrow Nov 10 '15 at 08:18
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    Yes but what I mean is that in algebra, it's not very clear that it's the same kind of "twisting". As far as I understand what's written in that link, it just means that the homology groups are "twisted" compared to what you would expect if you just knew the Betti numbers. – Najib Idrissi Nov 10 '15 at 08:22
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    @NajibIdrissi Since I don't know how to think of "twisted groups", I think of torsion in $H_1$ element-wise - a chain which isn't a boundary but becomes one after walking around it more than once. This fits in with my idea of twisting. Analogous intuition seems to work for $\pi _1$, except we look for contractibility. So I don't think it's the same kind of twisting, but it's similar enough to hope "twisting" could be made general enough to encompass all these notions. – Arrow Nov 10 '15 at 08:43

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It turns out that there is a relationship between analytic torsion and torsion in (co)homology. The idea is that analytic torsion equals Reidemeister torsion by the Cheeger-Muller theorem, and Reidemeister torsion is equal to the alternating product of sizes of torsion subgroups of integer homology (modulo some normalizing factors called regulators).

This relationship between the analytic and the algebraic is quite surprising! Unfortunately, it is not always well publicized in the analytic torsion literature (in my opinion), but it has gotten a lot of attention in recent research in number theory. It is explained (and applied) in, for example, this paper of Bergeron and Venkatesh: arXiv link.

To address your complete question, I don't know what hope there is of a "unified" theory of torsion. For example, I don't think that the torsion of a connection in differential geometry has anything to do with torsion of abelian groups.

Phillip Andreae
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