Perturbative invariants of 3 - manifolds with the first Betti number 1

It is known that perturbative invariants of rational homology 3-spheres can be formulated by using arithmetic perturbative expansion of quantum invariants of them. However, we could not make arithmetic perturbative expansion of quantum invariants for 3-manifolds with positive Betti numbers by the same method. In this paper, we explain how to make arithmetic perturbative expansion of quantum SO(3) invariants of 3-manifolds with the first Betti number 1. Further, motivated by this expansion, we formulate perturbative invariants of such 3-manifolds. We show some properties of the perturbative invariants, which imply that their coefficients are independent invariants. In the late 1980s, Witten [Wi] proposed topological invariants of a closed 3-manifold M for a simple compact Lie group G, what we call quantum G invariant, which is formally presented by a path integral whose Lagrangian is the Chern-Simons functional of G connections on M . There are two approaches to obtain mathematically rigorous information from a path integral: the operator formalism and the perturbative expansion. Motivated by the operator formalism of the Chern-Simons path integral, Reshetikhin and Turaev [ReT] gave the first rigorous mathematical construction of quantum invariants, as linear sums of quantum invariants of framed links. After that, rigorous constructions of quantum invariants were obtained by various approaches; in particular, Kirby and Melvin [KiM] formulated the quantum SO(3) invariant, which we denote by τ SO(3) r (M); it is defined to be a linear sum of the quantum sl2 invariant (the colored Jones polynomial) of framed links at an rth roots of unity. On the other hand, the perturbative expansion of the Chern-Simons path integral suggests that we can formulate perturbative invariants which describe asymptotic behavior of quantum invariants at r → ∞; in fact, it is known (see, e.g., [O3]) that we can formulate perturbative invariants of rational homology 3-spheres based on arithmetic perturbative expansion of quantum invariants of them. We review the construction of the perturbative SO(3) invariant of a rational homology 3-sphere M , as follows. Let p be an odd prime, and put ζ = exp(2π √ −1/p). Since it is known (by Murakami [M]) that τ SO(3) p (M) ∈ Z[ζ], we can make an expansion, τ p (M) = ap,0 + ap,1(ζ − 1) + ap,2(ζ − 1) + · · ·+ ap,N(ζ − 1) , with some integers ap,n’s. Though this expansion is not unique, (ap,n mod p) ∈ Z/pZ is uniquely determined by the value of τ SO(3) p (M), since Z[ζ] is isomorphic to Z[q]/T (q) where T (q) = q − 1 q − 1 = ( p 1 ) + ( p 2 ) (q − 1) + ( p 3 ) (q − 1) + · · ·+ ( p p ) (q − 1)p−1.