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

  • Vincent Florens

    Université de Pau et Université de Pau et des Pays de l'Adour, Pau, France
  • Gwénaël Massuyeau

    Université de Strasbourg, France

Abstract

Let be a field and let be a multiplicative subgroup. We consider the category of -dimensional cobordisms equipped with a representation of their fundamental group in , and the category of -linear maps defined up to multiplication by an element of . Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from to . In particular, when the group is free abelian and is the field of fractions of the group ring , we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for -manifolds with boundary; when 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 . 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.

Cite this article

Vincent Florens, Gwénaël Massuyeau, A functorial extension of the abelian Reidemeister torsions of three-manifolds. Enseign. Math. 61 (2015), no. 1/2, pp. 161–210

DOI 10.4171/LEM/61-1/2-8