Garoufalidis-Levine's finite type invariants for $\mathbb{Z}\pi$-homology equivalences from 3-manifolds to the 3-torus

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 $\pi ={\pi }_{1}N$-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 $N$.