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

TL;DR

This paper extends Fukaya's Morse homotopy invariant to 3-manifolds with first Betti number one by using a local system of rational functions, leading to new potential finite type invariants.

Contribution

It introduces a novel topological invariant for 3-manifolds with $b_1=1$ using Morse homotopy and rational function local systems, expanding Fukaya's framework.

Findings

Invariant takes values in Jacobi diagrams with rational function edges

Potential to produce many nontrivial finite type invariants

Connects Morse homotopy theory with rational function-based invariants

Abstract

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 homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.