Unified Quantum Invariants and Their Refinements for Homology 3–spheres with 2–torsion

For every rational homology 3–sphere with H1(M, Z) = (Z/2Z) 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. Introduction The Witten–Reshetikhin–Turaev (WRT) invariants of 3–manifolds, also known as quantum invariants, are defined only when the quantum parameter q is a certain root of unity. In [4], Habiro proposed a construction of a unified invariant of integral homology 3–spheres, dominating all quantum SU(2) invariants. The unified invariant is an element of the Habiro ring Ẑ[q] := lim ←−− n Z[q] (1− q)(1− q2)...(1− qn) . Every element f ∈ Ẑ[q] can be written as an infinite sum