Reidemeister torsion

Remarks

The class does not depend on the choice of $\delta$, and a change of the lifts or orientations of cells multiplies it by some $\pm \rho(g)$. It is invariant under subdivision and under simple homotopy equivalence. Example: for the circle with one $0$-cell and one $1$-cell, and $\rho(t) = \lambda \ne 1$ for a generator $t$ of $\pi = \Z$, the complex is $F \xrightarrow{\lambda - 1} F$, so $\Delta_\rho(S^1) = (\lambda - 1)^{\pm 1}$, depending on the convention. Non-example: for $\rho(t) = 1$, the twisted complex of the circle is $F \xrightarrow{0} F$, which is not acyclic, so the torsion is not defined. Conventions differ: some authors take the inverse, and some use representations $\pi \to GL_n(F)$ instead of ring homomorphisms to $F$. Reidemeister torsion is the key invariant in the classification of lens spaces: $L(p, q)$ and $L(p, q')$ are homeomorphic if and only if $q' \equiv \pm q^{\pm 1} \pmod p$.