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

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 $M$.

43 pages, 10 figures, v4, revised Theorem 1.2, Proposition 1.3 (supplement about orientability of fiberwise Morse function), Theorem 4.6 (AL-cycle must be replaced with the irreducible one), and made other few corrections

References in corpus (2)

• On the cube of the equivariant linking pairing for knots and 3-manifolds of rank one
• Statistics of Random 3-Manifolds occasionally fibering over the circle

Cited by in corpus (2)

• An invariant of fiberwise Morse functions on surface bundle over $S^1$ by counting graphs
• Finite type invariants of nullhomologous knots in 3-manifolds fibered over $S^1$ by counting graphs