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The relative cup-length in local Morse cohomology

Thomas O. Rot, Maciej Starostka, Nils Waterstraat

Abstract

Local Morse cohomology associates cohomology groups to isolating neighborhoods of gradient flows of Morse functions on (generally non-compact) Riemannian manifolds $M$. We show that local Morse cohomology is a module over the cohomology of the isolating neighborhood, which allows us to define a cup-length relative to the cohomology of the isolating neighborhood that gives a lower bound on the number of critical points of functions on $M$ that are not necessarily Morse. Finally, we illustrate by an example that this lower bound can indeed be stronger than the lower bound given by the absolute cup-length.

The relative cup-length in local Morse cohomology

Abstract

Local Morse cohomology associates cohomology groups to isolating neighborhoods of gradient flows of Morse functions on (generally non-compact) Riemannian manifolds . We show that local Morse cohomology is a module over the cohomology of the isolating neighborhood, which allows us to define a cup-length relative to the cohomology of the isolating neighborhood that gives a lower bound on the number of critical points of functions on that are not necessarily Morse. Finally, we illustrate by an example that this lower bound can indeed be stronger than the lower bound given by the absolute cup-length.
Paper Structure (7 sections, 3 theorems, 25 equations, 3 figures)

This paper contains 7 sections, 3 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Local Morse cohomology
  3. Cup product in Morse cohomology
  4. Cup products in local Morse Cohomology
  5. The relative cup length in local Morse cohomology
  6. Example
  7. Appendix: Orientations and multiple intersections.

Key Result

Lemma 2.2

Suppose $N$ is an isolating neighborhood of flows $\varphi^{\alpha},\varphi^{\beta}$, and $\varphi^\gamma$. Suppose that $\varphi^{\alpha}=\varphi^\gamma$ or $\varphi^{\beta}=\varphi^\gamma$. Then the flows are isolation compatible.

Figures (3)

  • Figure 1: The one-dimensional moduli space $W(z,x,y)$ (in blue) is typically not compact. The non-compact ends are parametrized by the three types of breaking that can occur. Equation \ref{['eq:cup']} expresses that the moduli space can be compactified to a one dimensional manifold with boundary by adjoining these possible breakings at the non-compact ends.
  • Figure 2: A sketch of the main idea of the proof of Theorem \ref{['thm:main']}. A non-zero relative cup length for every $j$ means that there exist non-vacuous diagrams in $N$ as in the figure (here for $n=3$). The points $x_1^j$ up to $x_n^j$ converge to (different) critical points $x_1,\ldots x_n$ of $f^\alpha$ as $j\rightarrow \infty$.
  • Figure 3: On the left is a sketch of the gradient flow of $f(x)=\frac{1}{2}\langle x, R x \rangle$ on the two sphere depicted. On the right the gradient flow of the projectivization of two such Morse functions on $\mathbb{R}\mathbb{P}^2$ is shown. The intersection $W^u([p_2^\alpha] ;\mathcal{Q}^\alpha)\cap W^s([p_1^\alpha];\mathcal{Q}^\alpha)\cap W^s([p_1^\beta];\mathcal{Q}^\beta)$consists of a single point drawn in purple. This shows that $\eta^{[p_1^\alpha]}\smallsmile \eta^{[p_1^\beta]}=\eta^{[p_2^\alpha]}.$

Theorems & Definitions (10)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4