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Morse-Novikov homology and $β$-critical points

Adrien Currier

TL;DR

This work develops a Morse theory for $\beta$-critical points on a closed manifold using Morse-Novikov homology $HN_*(M,\beta)$ to obtain lower bounds on the number of $\beta$-critical points of a $\beta$-Morse function $f$ and extends the bounds to generating functions that are quadratic at infinity. The central technique combines the Morse-Novikov chain complex, integral covers, and cobordism-based arguments to relate critical-point counts to $HN_*(M,\beta)$, yielding $\# Crit_i^\beta(f) \ge \mathrm{rank}(HN_i(M,\beta))$ and, for generating functions, $\# Crit_j^\beta(F) \ge \mathrm{rank}(HN_{j-p}(M,\beta))$ with $p$ the negative-definite dimension at infinity. The paper then applies these ideas to locally conformally symplectic geometry by bounding the number of essential Liouville chords between Lagrangians via Morse-Novikov ranks, and it discusses a path toward a derived-sheaf perspective for $\frak{lcs}$ topology. Overall, the results provide a robust link between Morse-Novikov invariants and geometric counting problems in $\frak{lcs}$-type settings. $HN_*(M,\beta)$ serves as the main invariant governing the minimal number of distinguished critical points and chords under generic Morse conditions.

Abstract

Given a manifold $M$, some closed $β\inΩ^1(M)$ and a map $f\in C^\infty(M)$, a $β$-critical point is some $x\in M$ such that $d_βf_{x}=0$ for the Lichnerowicz derivative $d_β$. In this paper, we will give a lower bound for the number of $β$-critical points of index $i$ of a $β$-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.

Morse-Novikov homology and $β$-critical points

TL;DR

This work develops a Morse theory for -critical points on a closed manifold using Morse-Novikov homology to obtain lower bounds on the number of -critical points of a -Morse function and extends the bounds to generating functions that are quadratic at infinity. The central technique combines the Morse-Novikov chain complex, integral covers, and cobordism-based arguments to relate critical-point counts to , yielding and, for generating functions, with the negative-definite dimension at infinity. The paper then applies these ideas to locally conformally symplectic geometry by bounding the number of essential Liouville chords between Lagrangians via Morse-Novikov ranks, and it discusses a path toward a derived-sheaf perspective for topology. Overall, the results provide a robust link between Morse-Novikov invariants and geometric counting problems in -type settings. serves as the main invariant governing the minimal number of distinguished critical points and chords under generic Morse conditions.

Abstract

Given a manifold , some closed and a map , a -critical point is some such that for the Lichnerowicz derivative . In this paper, we will give a lower bound for the number of -critical points of index of a -Morse function 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.

Paper Structure

This paper contains 17 sections, 16 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Morse-Novikov homology
  4. A Morse theory for $\beta$-critical points
  5. A couple of inequalities of ranks
  6. Morse theory for cobordisms between manifolds with boundary
  7. On the lift of $\beta$-Morse functions
  8. Extension 1.
  9. Extension 2.
  10. Extension 3.
  11. Extension 4.
  12. Final observation.
  13. A lower bound for $\# Crit_\beta(f)$
  14. A generalization to generating functions
  15. (Essential) Liouville chords
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $M$ be a closed connected manifold and $\beta\in\Omega^1(M)$ be closed. We will call $\beta$ the various pullbacks of $\beta$. Take $F:M\times\mathbb{R}^m\rightarrow\mathbb{R}$ (for some $m\in\mathbb{N}$) a smooth map that is equal to a quadratic form outside of a compact. Let $p$ be the dimensi where $HN$ stands for Morse-Novikov homology.

Figures (3)

  • Figure 1: Example of $V$ for a genus $2$ surface.
  • Figure 2: Example of $W_0^-$, $W_1^-$ and $W_2^-$.
  • Figure :

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Definition 1.3
  • Proposition 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • ...and 38 more