A unified quantum SO (3) invariant for rational homology 3-spheres

Given a rational homology 3-sphere M with |H 1(M,ℤ)|=b and a link L insideM, colored by odd numbers, we construct a unified invariant I M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L=∅, I M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined byI M. Important applications are the new Ohtsuki series (perturbative expansions ofI M) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invarian