Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

TL;DR

This paper establishes a relationship between Morse theory, Reidemeister torsion, and Seiberg-Witten invariants for 3-manifolds, proposing a conjecture linking these invariants and torsion in a new geometric framework.

Contribution

It introduces a formula connecting gradient flow orbits, Morse complex torsion, and Reidemeister torsion, and proposes a conjecture relating Seiberg-Witten invariants to Milnor torsion in 3-manifolds.

Findings

Derived a formula relating flow orbits, Morse torsion, and Reidemeister torsion.

Proposed a conjecture linking Seiberg-Witten invariants with Milnor torsion.

Deduced the Meng-Taubes relation from the conjecture.

Abstract

Let X be a compact oriented Riemannian manifold and let $\phi:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $\phi$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $\phi$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $\phi$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $\phi$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X.