Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion

Anna Beliakova ¹   ; Christian Blanchet ²   ; Thang T. Q. Lê ³

¹ Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich, Switzerland
² Institut de Mathématiques de Jussieu (UMR-CNRS 7586) Université Paris Diderot 175 rue du Chevaleret F-75013 Paris, France
³ School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160, U.S.A.

Fundamenta Mathematicae, Tome 201 (2008) no. 3, pp. 217-239

Résumé

For every rational homology $3$-sphere with $H_1(M,\mathbb Z) =(\mathbb Z/2\mathbb Z)^n$ we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring) such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten–Reshetikhin–Turaev invariant at this root, and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction.

Détail

DOI : 10.4064/fm201-3-2

Keywords: every rational homology sphere mathbb mathbb mathbb construct unified invariant which takes values certain cyclotomic completion polynomial ring evaluation invariant odd root unity provides witten reshetikhin turaev invariant root even root unity quantum invariant moreover unified invariant splits sum refined unified invariants dominating spin cohomological refinements quantum invariants results ohtsuki series integrality quantum invariants main applications construction