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It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by "(semi-)classical" invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant. In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the m reduction at large m limit of the ℓ -loop part of the refinement of the LMO invariant for ℓ ≤ 5 , we formulate an ℓ -variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant. [ABSTRACT FROM AUTHOR]

In our earlier work, we studied the Z ^ -invariant(or homological blocks) for SO(3) gauge group and we found it to be same as Z ^ S U (2) . This motivated us to study the Z ^ -invariant for quotient groups S U (N) / Z m , where m is some divisor of N. Interestingly, we find that Z ^ -invariant is independent of m. [ABSTRACT FROM AUTHOR]