A refinement of the LMO invariant for 3-manifolds with the first Betti number 1 (Q6543114)

In quantum topology, \textit{T. Le}, \textit{J. Murakami} and the present author [Topology 37, No. 3, 539--574 (1998; Zbl 0897.57017)] introduced the LMO invariant for closed \(3\)-manifolds, which is an infinite linear sum of trivalent graphs. For integral homology \(3\)-spheres, the LMO invariant dominates perturbative invariants and finite type invariants, and hence, quantum invariants. Furthermore, the LMO invariant is expected to classify integral homology \(3\)-spheres. However, it is known that the LMO invariant is not a complete invariant for rational homology \(3\)-spheres, and it is relatively weak for \(3\)-manifolds with \(b_1>0\), where \(b_1\) denotes the first Betti number.\N\NIn the article under review, the author constructs a refinement of the LMO invariant for \(3\)-manifolds with \(b_1=1\) (Theorems~1.1 and 1.2). He also defines the \(2\)-loop polynomial of such \(3\)-manifolds as the \(2\)-loop part of the refinement of the LMO invariant, which serves as a refinement of the Casson-Walker-Lescop invariant and relates to the \(2\)-loop polynomial for knots introduced by the author [Geom. Topol. 11, 1357--1475 (2007; Zbl 1154.57012)].\N\NFor a \(3\)-manifold \(M\) with \(b_1(M)=1\), the refinement \(Z(M)\) is defined to be a certain equivalence class of the loop expansion of the LMO invariant of a pair \((N,K)\), where \(N\) is a rational homology \(3\)-sphere and \(K\) is a null-homologous \(0\)-framed knot in \(N\) such that \(M\) is obtained from \(N\) by Dehn surgery on \(K\). This construction follows the spirit of \(\mathbb{Z}\)-equivariant invariants, cf. [\textit{C. Lescop}, ``On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one'', Preprint, \url{arXiv:1008.5026}; \textit{T. Ohtsuki}, Ser. Knots Everything 40, 253--262 (2007; Zbl 1161.57006); \textit{T. Watanabe}, ``Morse theory and Lescop's equivariant propagator for 3-manifolds with $b_1=1$ fibered over $S^1$'', Preprint, \url{arXiv:1403.8030}].