Morse theory and Lescop's equivariant propagator for 3-manifolds with $b_1=1$ fibered over $S^1$

TL;DR

This paper introduces AL-paths on fibered 3-manifolds with $b_1=1$, linking Morse theory to Lescop's propagator and enabling a $bZ$-equivariant Chern--Simons theory.

Contribution

It defines AL-paths and relates their moduli space to Lescop's equivariant propagator, advancing the understanding of 3-manifold invariants.

Findings

AL-paths encode the Lefschetz zeta function of the manifold.

The moduli space of AL-paths explicitly constructs Lescop's equivariant propagator.

Provides a framework for $bZ$-equivariant Chern--Simons perturbation theory.

Abstract

For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the fiberwise gradient $\xi$ of a fiberwise Morse function on $M$, we introduce the notion of amidakuji-like path (AL-path) on $M$. An AL-path is a piecewise smooth path on $M$ consisting of edges each of which is either a part of a critical locus of $\xi$ or a flow line of $-\xi$. Counting closed AL-paths with signs gives the Lefschetz zeta function of $M$. The "moduli space" of AL-paths on $M$ gives explicitly Lescop's equivariant propagator, which can be used to define $\mathbb{Z}$-equivariant version of Chern--Simons perturbation theory for $M$.