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<p class="noindent">It is known that perturbative invariants of rational homology 3–spheres can be constructed 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. </p>

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