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In this paper, it is explained that a topological invariant for 3-manifold$M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopytheoretic approach for Chern--Simons perturbation theory to a local system on$M$ of rational functions associated to the free abelian covering of $M$. Ourinvariant takes values in Garoufalidis--Rozansky's space of Jacobi diagramswhose edges are colored by rational functions. It is expected that ourinvariant gives a lot of nontrivial finite type invariants of 3-manifolds.

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with $b_1=1$