Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion
Volume 201 / 2008
Fundamenta Mathematicae 201 (2008), 217-239 MSC: Primary 57N10; Secondary 57M25 DOI: 10.4064/fm201-3-2
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.