A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let $\mathbb{F}$ be a field and let $G\subset \mathbb{F}\setminus \{0\}$ be a multiplicative subgroup. We consider the category $\mathcal{Cob}_G$ of $3$-dimensional cobordisms equipped with a representation of their fundamental group in $G$, and the category ${Vect}_{\mathbb{F},\pm G}$ of $\mathbb{F}$-linear maps defined up to multiplication by an element of $\pm G$. Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from $Cob_G$ to ${Vect}_{\mathbb{F},\pm G}$. In particular, when the group $G$ is free abelian and $\mathbb{F}$ is the field of fractions of the group ring $\mathbb{Z}[G]$, we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for $3$-manifolds with boundary; when $G$ is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup $G\subset \mathbb{F}\setminus \{0\}$. We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.