The aim of this note is to introduce -Alexander torsions for 3-manifolds (which are generalizations of the usual Alexander polynomial and also of the -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 -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 (généralisations du polynôme d'Alexander usuel et de l'invariant d'Alexander 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 permettent de détecter la norme de Thurston d'une variété de dimension 3 irréductible et qu'elles sont symétriques.
Accepted:
Published online:
Jérôme Dubois 1; Stefan Friedl 2; Wolfgang Lück 3
@article{CRMATH_2015__353_1_69_0, author = {J\'er\^ome Dubois and Stefan Friedl and Wolfgang L\"uck}, title = {The $ {L}^{2}${-Alexander} torsions of 3-manifolds}, journal = {Comptes Rendus. Math\'ematique}, pages = {69--73}, publisher = {Elsevier}, volume = {353}, number = {1}, year = {2015}, doi = {10.1016/j.crma.2014.10.012}, language = {en}, }
Jérôme Dubois; Stefan Friedl; Wolfgang Lück. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.012/
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