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.



Is there a geometric interpretation for Reidemeister torsion?
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*.


