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

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.