  Garoufalidis and Levine defined a filtration for 3-manifolds equipped with
some degree 1 map ($\mathbb{Z}\pi$-homology equivalence) to a fixed 3-manifold
$N$ and showed that there is a natural surjection from a space of
$\pi=\pi_1N$-decorated graphs to the graded quotient of the filtration over
$\mathbb{Z}[\frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$
the surjection of Garoufalidis--Levine is actually an isomorphism over
$\mathbb{Q}$. For the proof, we construct a perturbative invariant by applying
Fukaya's Morse homotopy theoretic construction to a local system of the
quotient field of $\mathbb{Q}\pi$. The first invariant is an extension of the
Casson invariant to $\mathbb{Z}\pi$-homology equivalences to the 3-torus. The
results of this paper suggest that there is a highly nontrivial equivariant
quantum invariants for 3-manifolds with $b_1=3$. We also discuss some
generalizations of the perturbative invariant for other target spaces $N$.
