Morse-Novikov homology and $\beta$-critical points

TL;DR

This paper establishes lower bounds on the number of $eta$-critical points of Morse functions using Morse-Novikov homology, extends results to generating functions, and applies findings to detect essential Liouville chords in symplectic geometry.

Contribution

It introduces a lower bound for $eta$-critical points based on Morse-Novikov homology and generalizes to generating functions, with applications in symplectic geometry.

Findings

Lower bounds for $eta$-critical points in terms of Morse-Novikov homology

Extension of results to generating functions quadratic at infinity

Application to detecting essential Liouville chords

Abstract

Given a manifold $M$, some closed $\beta\in\Omega^1(M)$ and a map $f\in C^\infty(M)$, a $\beta$-critical point is some $x\in M$ such that $d_\beta f_{x}=0$ for the Lichnerowicz derivative $d_\beta$. In this paper, we will give a lower bound for the number of $\beta$-critical points of index $i$ of a $\beta$-Morse function $f$ in terms of the Morse-Novikov homology, and we generalize this result to generating functions (quadratic at infinity). We also give an application to the detection of essential Liouville chords of a set length. These are a type of chords that appear in locally conformally symplectic geometry as even-dimensional analogues to Reeb chords.