On a twisted Reidemeister torsion (Q2790211)

The Reidemeister torsions of a cell complex are a family of invariants arising from linear representations of its fundamental group. They are usually defined modulo certain choices, but under some conditions on the representation these dependencies can disappear. The setting of the present paper, originating from theoretical physics [\textit{V. Mathai} and \textit{S. Wu}, J. Differ. Geom. 88, No. 2, 297--332 (2011; Zbl 1238.58023)], is the following: let \(K\) be a finite CW-complex and \(\rho: \pi_1(K) \to \mathrm{GL}(E)\) a complex linear representation. Then the choice of a Euclidean metric on \(E\) determines a Reidemeister torsion \(\tau(K, E)\), an element of the determinant line of the cohomology, i.e. the 1-dimensional complex vector space \(\det(H^\bullet(K, E))\) (the alternating tensor product of the determinant lines of the cohomology spaces \(H^i(K, E)\)). The goal of this short note is to produce a twisted version of this torsion, depending additionally on an odd cocycle. To do this, instead of the classical cochain complex the author uses a complex of currents invented by \textit{J. L. Dupont} [Lect. Notes Math. 1318, 87--91 (1988; Zbl 0677.55016)]. The additional parameter \(t\) is then a closed element of odd degree in this complex and the resulting torsion is denoted by \(\tau_{\mathrm{twist}}(K, E, t)\). A similar definition was made by Mathai and Wu [loc. cit.], depending on an odd-degree cocycle \(\vartheta\) on \(K\) with coefficients in \(E\) of sufficiently large degree (larger than half \(\dim K\)); the author checks that his definition agrees with the former in the relevant cases.