2.3 Local behavior near critical points

Critical points are key to understanding how functions behave. They're where the function's gradient is zero, and they tell us a lot about the function's shape and properties. This section dives into what happens near these special points.

We'll learn about the Morse Lemma, which gives us a simple way to describe functions near critical points. This helps us classify critical points and understand how the function changes around them. It's super useful for analyzing functions in many areas of math.

Level Sets and Sublevel Sets

Topological Structures Defined by Functions

• Level sets consist of all points in the domain of a function where the function takes on a specific value
• Sublevel sets contain all points in the domain where the function value is less than or equal to a given value
• Level sets and sublevel sets provide a way to analyze the topology of a function's domain based on its values
• As the value defining a sublevel set increases, the sublevel set grows, potentially undergoing topological changes (adding or removing connected components, holes, or other features)

Topological Changes and Critical Points

• Topological changes in sublevel sets occur at critical values of the function, where the gradient vanishes
• Critical points are points in the domain where the gradient of the function is zero
• The behavior of a function near its critical points determines the types of topological changes that occur in the sublevel sets
• Morse theory studies the relationship between critical points and the topology of level sets and sublevel sets

Morse Functions and Charts

Properties of Morse Functions

• Morse functions are smooth functions whose critical points are non-degenerate (the Hessian matrix at the critical point has full rank)
• Non-degenerate critical points are isolated, meaning there are no other critical points in a small neighborhood around them
• The Morse Lemma states that near a non-degenerate critical point, a Morse function can be locally expressed as a quadratic form (sum of squares of coordinates with coefficients ±1)
• Morse functions are dense in the space of smooth functions, meaning that any smooth function can be approximated by a Morse function

Morse Charts and Morse-Bott Functions

• Morse charts are local coordinate systems around a critical point in which the Morse function takes the form given by the Morse Lemma
• In a Morse chart, the index of a critical point is the number of negative coefficients in the quadratic form
• Morse-Bott functions are a generalization of Morse functions that allow for critical submanifolds (sets of critical points forming a submanifold) instead of isolated critical points
• The Morse-Bott Lemma provides a local form for Morse-Bott functions near critical submanifolds, generalizing the Morse Lemma

Morse Inequalities and Topology

• Morse inequalities relate the number of critical points of each index to the Betti numbers (ranks of homology groups) of the manifold
• The weak Morse inequalities state that the number of critical points of index $k$ is greater than or equal to the $k$-th Betti number
• The strong Morse inequalities provide more refined relationships between critical points and Betti numbers, involving alternating sums
• Morse inequalities can be used to deduce topological properties of a manifold from the critical points of a Morse function defined on it

Stable and Unstable Manifolds

Dynamics Near Critical Points

• Stable and unstable manifolds are important objects in the study of the local behavior of a dynamical system near a critical point
• The stable manifold of a critical point consists of all points whose forward trajectories converge to the critical point
• The unstable manifold of a critical point consists of all points whose backward trajectories converge to the critical point
• The dimensions of the stable and unstable manifolds are related to the index of the critical point (number of negative eigenvalues of the Hessian)

Gradient Flows and Morse Homology

• In the context of Morse theory, stable and unstable manifolds are studied for the gradient flow of a Morse function
• The gradient flow is a vector field on the manifold that points in the direction of steepest ascent of the Morse function
• Integral curves of the gradient flow are paths that follow the direction of steepest ascent, connecting critical points
• Stable and unstable manifolds of critical points intersect transversely, forming the basis for the construction of Morse homology
• Morse homology is a powerful tool for studying the topology of a manifold using the critical points and gradient flow of a Morse function defined on it