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Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the Reidemeister-Franz torsion $\tau^\varphi(X)\in F$ is an invariant of the twisted chain complex, well-defined up to multiplication by $\pm\varphi(\pi_1(X))$. (The value can be pinned down further by choosing a "homology orientation" and an "Euler structure".)

The definition of the torsion is rather opaque to me; it involves products of determinants of basis-change matrices. I vaguely see that it has something to do with how well a splitting of the acyclic chain complex respects the cellular basis.

My question is whether there is a geometric interpretation of Reidemeister-Franz torsion. For instance, what is it measuring about the space? Is there an object in $X$ representing the torsion? Feel free to restrict the category if it helps.

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Yes, there is a very nice geometric interpretation over any manifold $X$ satisfying $\chi(X)=0$, for a version of Reidemeister torsion spelled out in Turaev's paper "Euler structures, nonsingular vector fields, and torsions of Reidemeister type". Ian Agol's comment mentioned the Seiberg-Witten invariants, which equals this version of torsion in 3 dimensions. It turns out that both of these are equal to an $S^1$-valued Morse theoretic invariant $I(X)$ defined by Hutchings and Lee in their PhD work:

"Circle-valued Morse theory and Reidemeister torsion" (Geom. Top. Vol.3, 1999)

Roughly speaking, we equip $X$ with a suitable Morse function $f:X\to S^1$ for which all critical points and closed orbits of $-\nabla f$ are nondegenerate. Then $I(X)$ is defined by suitably counting the Morse-flowlines between critical points as well as the closed periodic orbits (remembering their periods). The invariant is independent of the choice of $f$ and it is identified with the Redeimester torsion (for a suitable homology orientation). When $\dim X=3$ and $b_1(X)>0$, this also recovers the Seiberg-Witten invariants. We can view $I(X)$ as the analog of the Gromov invariant in 4 dimensions which suitably counts $J$-holomorphic curves.

Here is a simple example which may make the "involvement of products of determinants of basis-change matrices" less opaque to you. When $f:X\to S^1$ is a fiber bundle with fiber $\Sigma$, the periodic flow of $-\nabla f$ defines a self-map $\varphi:\Sigma\to\Sigma$, and $I(X)$ effectively counts fixed points of $\varphi^k$ weighted by their Lefschetz-sign. The identification with torsion translates into the Lefschetz fixed-point theorem.

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  • $\begingroup$ The interpretation as a fixed-point theorem for a return map is interesting; thanks for giving the example. It will take me a while to digest this answer, but I'll mark it as accepted for now because it is more than what I was hoping to see. $\endgroup$
    – Kyle Miller
    Oct 16, 2017 at 18:54
  • $\begingroup$ Are there analogues of Turaev's results in the case that $\chi(X)$ is non-trivial? $\endgroup$
    – Tim Porter
    Oct 22, 2017 at 6:26
  • $\begingroup$ I do not know. Though for $\chi(X)$ zero or nonzero in dimension 4 there is an analog of $I(X)$, it’s the Gromov invariant which counts $J$-holomorphic curves (defined for symplectic manifolds by Taubes, and for non-symplectic manifolds by myself in my PhD thesis). $\endgroup$
    – Chris Gerig
    Oct 22, 2017 at 8:22
  • $\begingroup$ The result mentioned above is really due to Milnor. Suppose that $f: X \to X$ is a self-map of a space which has the homotopy type of a finite complex. Then the Lefshetz zeta function $L_f(t) = \text{exp}(\sum L(f^n)t^n)$ and it gives a homological count of the periodic orbits of $f$ (which is the same as the periodic orbits of the associated dynamical system on the mapping torus $M_f$ of $f$. Milnor showed that this zeta function and the Reidemeister torsion wrt the representation of the fundamental group into the field of rational functions on one variable coincide after a minor change. $\endgroup$
    – John Klein
    Oct 24, 2017 at 13:13
  • $\begingroup$ @JohnKlein Yes that is a good clarification of due credit: in this example with no critical points it's precisely Milnor's invariant. $\endgroup$
    – Chris Gerig
    Oct 24, 2017 at 22:14
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Have you looked at the relation with Whitehead torsion and simple homotopy theory? I forget the details and am no expert, but there is a lot of nice low dimensional geometric topology hidden in that theory if that is any help. Look at Milnor's paper, Bull. Amer. Math. Soc., 72 (3):(1966) 358–426. but also look at the Wikipedia articles on both types of torsion for a quick entry point to the ideas.

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  • $\begingroup$ I suppose I should read that Milnor paper more carefully since everyone seems to recommend it :-) Section 9, about smooth manifolds, seems still rather algebraic, but I can believe I read through it too quickly to be able to "see" torsions. (Regarding your earlier suggestion about K-theory, I actually asked a budding algebraic K-theorist about this at some point, and he started describing what he knew about $K_1$ to me, though we couldn't see what it had to do with the original space.) $\endgroup$
    – Kyle Miller
    Oct 16, 2017 at 19:09
  • $\begingroup$ There was a nice paper by Eckmann and a paper by Larry Siebenmann which gave a purely geometric approach to Whitehead torsion. (I worked out an abstract version of this in my book with Heiner Kamps.) There is sense in which it all comes down to a generalisation of rewriting theory. You know about manipulating knots and how that can be reflected in the group presentations of the knot group. Reidemeister developed both the geometric and some of the algebraic side extending that sort of combinatorial theory. Whitehead's Combinatorial Homotopy took that further. (out of characters!) $\endgroup$
    – Tim Porter
    Oct 16, 2017 at 19:44
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    $\begingroup$ Whitehead's simple homotopy theory is designed to see what the obstructions to doing manipulations on the cell structure / handle decomposition of a Cw-complex / manifold to see if you can get all of the similar spaces in the homotopy type ... and you cannot. The obstruction is the Whitehead torsion. So a naive approach to modelling homotopy type combinatorially fails, but gives birth to algebraic K-theory via the Whitehead group of the fundamental group. $\endgroup$
    – Tim Porter
    Oct 16, 2017 at 19:48

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