Turaev torsion

Turaev torsion is a sign-refined form of Reidemeister torsion for manifolds and chain complexes. For a knot exterior with meridian $\mu$ and first Betti number one, its group-ring normalization satisfies

The refinement resolves sign and normalization choices present in Reidemeister torsion.

A homology orientation chooses an orientation of the real vector space ${⨁}_i{H}_i(M;\mathbb{R})$. It fixes the sign in refined Turaev torsion.

For a connected finite CW complex $X$ with Euler characteristic zero, choose a point $p_e$ in each cell. An Euler chain is a singular chain $\xi$ of degree one satisfyingWith these points fixed, two Euler chains define the same Euler structure when their difference is a chain boundary. Adding cycles makes their classes a set with a simply transitive group action of $H_1(X;\mathbb{Z})$. This choice fixes the cellular-lift ambiguity in Turaev torsion; a homology orientation fixes its sign.^[ref]

For a connected finite CW complex $X$, choose a point $p_e$ in each open cell $e$. An Euler chain is a singular chain $\xi$ of degree one withIt exists exactly when the Euler characteristic of $X$ is zero. Necessity follows because the total coefficient sum of every one-chain boundary is zero, whereas the displayed zero-chain has coefficient sum $\chi (X)$.

For sufficiency, choose a base point $b$ and paths ${\gamma }_e$ from $b$ to $p_e$. Then $\xi ={\sum }_e(-1{)}^{\dim e}{\gamma }_e$ has boundary equal to the required sum minus $\chi (X)b$, so it works when $\chi (X)=0$. Connected CW complexes are path connected, which justifies choosing these paths.

With the cell points fixed, two Euler chains differ by a cycle and define the same Euler structure when that cycle is a boundary. Adding homology classes makes Euler structures a torsor for $H_1(X;\mathbb{Z})$. This extra choice removes an ambiguity in Turaev torsion.^[ref]

Let $Y$ be obtained by Dehn filling a torus boundary of $X$, with core class $h\in H_1(Y)$. Choose a one-dimensional coefficient representation $\rho :H_1(Y)\to F^{\times }$ over a field, with $\rho (h)\ne 1$, and assume the associated twisted chain complexes are acyclic. Then, up to the usual torsion unit,With compatible Euler structures and homology orientations, the refined formula fixes that ambiguity. The denominator comes from the attached solid torus's circle complex. When $\rho (h)=1$, this division is unavailable; additional homology data are needed.^[ref]