The $ {L}^{2}$-Alexander torsions of 3-manifolds

Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang

doi:10.1016/j.crma.2014.10.012

Geometry/Topology

The

L^{2}

-Alexander torsions of 3-manifolds

Presented by: Étienne Ghys

Dubois, Jérôme ¹; Friedl, Stefan ²; Lück, Wolfgang ³

¹Université Blaise Pascal – Laboratoire de Mathématiques, UMR 6620 – CNRS, Campus des Cézeaux – B.P. 80026, 63171 Aubière cedex, France
²Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
³Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 69-73

Abstract (Original language)
Résumé

The aim of this note is to introduce $L^{2}$ -Alexander torsions for 3-manifolds (which are generalizations of the usual Alexander polynomial and also of the $L^{2}$ -Alexander invariant defined by Li and Zhang [7]) and to report on calculations for graph manifolds and fibered 3-manifolds. We further announce that given any irreducible 3-manifold, there exists a coefficient system such that the corresponding $L^{2}$ -Alexander torsion detects the Thurston norm. Finally we also state a symmetry formula.

Le but de cette note est d'introduire les torsions d'Alexander $L^{2}$ (généralisations du polynôme d'Alexander usuel et de l'invariant d'Alexander $L^{2}$ défini par Li et Zhang [7]) et d'en donner le calcul pour les variétés graphées et les variétés fibrées de dimension 3. On annonce enfin que les torsions d'Alexander $L^{2}$ permettent de détecter la norme de Thurston d'une variété de dimension 3 irréductible et qu'elles sont symétriques.

Received: 2014-08-19
Accepted: 2014-10-16
Published online: 2014-11-04

DOI: 10.1016/j.crma.2014.10.012

Author's affiliations:

Dubois, Jérôme ¹ ; Friedl, Stefan ² ; Lück, Wolfgang ³

¹ Université Blaise Pascal – Laboratoire de Mathématiques, UMR 6620 – CNRS, Campus des Cézeaux – B.P. 80026, 63171 Aubière cedex, France
² Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
³ Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

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     title = {The $ {L}^{2}${-Alexander} torsions of 3-manifolds},
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Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang. The $ {L}^{2}$-Alexander torsions of 3-manifolds. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 69-73. doi: 10.1016/j.crma.2014.10.012

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