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

Abstract

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 $N$.

The statement of the main theorem (Theorem 1.1 in Section 1) needs to be modified due to an error. The correct statement is weaker: the "isomorphism" is in fact an epimorphism onto a space of graphs. We are working with coauthors on other examples for which this problem does not occur

References in corpus (3)

• On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres
• On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one
• Kirby calculus for null-homologous framed links in 3-manifolds

Cited by in corpus (1)

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