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

Volume 201 / 2008

Anna Beliakova, Christian Blanchet, Thang T. Q. Lê Fundamenta Mathematicae 201 (2008), 217-239 MSC: Primary 57N10; Secondary 57M25 DOI: 10.4064/fm201-3-2

Abstract

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.

Authors

  • Anna BeliakovaInstitut für Mathematik
    Universität Zürich
    Winterthurerstrasse 190
    CH-8057 Zürich, Switzerland
    e-mail
  • Christian BlanchetInstitut de Mathématiques de Jussieu
    (UMR-CNRS 7586)
    Université Paris Diderot
    175 rue du Chevaleret
    F-75013 Paris, France
    e-mail
  • Thang T. Q. LêSchool of Mathematics
    Georgia Institute of Technology
    Atlanta, GA 30332-0160, U.S.A.
    e-mail

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