🔁Elementary Differential Topology Unit 15 – Morse Theory: Fundamentals and Applications

Key Concepts and Definitions

• Morse theory studies the relationship between the topology of a manifold and the critical points of a smooth function defined on it
• A smooth function $f: M \to \mathbb{R}$ is a Morse function if all its critical points are non-degenerate
• Critical points of a Morse function are points where the gradient $\nabla f$ vanishes
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • The Hessian matrix is the matrix of second partial derivatives of $f$
• The Morse index theorem relates the index of a critical point to the topology of the sublevel sets $f^{-1}((-\infty, a])$
• The Euler characteristic of a compact manifold can be computed using the Morse inequalities, which relate the number of critical points of each index to the Betti numbers of the manifold

Historical Context and Development

• Marston Morse developed the theory in the 1920s and 1930s, building on earlier work by Poincaré and Birkhoff
• Morse's original motivation was to study the topology of loop spaces in variational calculus
• In the 1950s and 1960s, Smale and others further developed the theory and explored its connections to dynamical systems and differential topology
• The Morse homology theory, introduced by Witten in the 1980s, provides a powerful algebraic framework for studying Morse theory
  • Morse homology is related to other homology theories, such as singular homology and Floer homology
• Morse theory has found applications in diverse areas of mathematics, including algebraic topology, differential geometry, and mathematical physics

Mathematical Foundations

• Morse theory relies on the machinery of smooth manifolds and differential topology
• The implicit function theorem and the inverse function theorem are key tools for studying the local behavior of smooth functions
• The Morse lemma states that near a non-degenerate critical point, a Morse function can be expressed in a canonical form using a suitable coordinate system
• Morse functions are dense in the space of smooth functions, meaning that any smooth function can be approximated by a Morse function
  • This density result is crucial for applying Morse theory to the study of arbitrary smooth manifolds
• The gradient flow of a Morse function provides a way to decompose a manifold into elementary pieces called handles
• Morse theory can be generalized to study functions on infinite-dimensional manifolds, such as the energy functional in the calculus of variations

Morse Functions and Critical Points

• A critical point of a smooth function $f: M \to \mathbb{R}$ is a point where the gradient $\nabla f$ vanishes
• A critical point is non-degenerate if the Hessian matrix at that point is non-singular
  • The Hessian matrix encodes information about the second-order behavior of $f$ near the critical point
• The index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian matrix
• Morse functions are characterized by having only non-degenerate critical points
• The critical points of a Morse function provide a way to decompose the manifold into elementary pieces called handles
• The arrangement of critical points and their indices determine the topology of the manifold
  • For example, on a surface, minima correspond to 0-handles, saddles to 1-handles, and maxima to 2-handles

The Morse Lemma and Local Structure

• The Morse lemma states that near a non-degenerate critical point $p$ of index $\lambda$, a Morse function $f$ can be expressed in the canonical form $f(x) = f(p) - x_1^2 - \cdots - x_\lambda^2 + x_{\lambda+1}^2 + \cdots + x_n^2$
  • This canonical form is achieved by a suitable choice of local coordinates near $p$
• The Morse lemma provides a standard model for the local behavior of a Morse function near its critical points
• The sublevel sets $f^{-1}((-\infty, a])$ undergo a topological change when $a$ passes through a critical value
  • This change is described by attaching a handle of index $\lambda$ to the sublevel set
• The Morse lemma implies that the sublevel sets are locally homeomorphic to a standard model near each critical point
• The proof of the Morse lemma relies on the implicit function theorem and the inverse function theorem from multivariable calculus

Handlebody Decomposition

• Morse functions provide a way to decompose a manifold into elementary pieces called handles
• A handle of index $\lambda$ is a product of disks $D^\lambda \times D^{n-\lambda}$ attached to the boundary of a manifold along $\partial D^\lambda \times D^{n-\lambda}$
  • The attaching map is determined by the gradient flow of the Morse function near a critical point of index $\lambda$
• The handlebody decomposition of a manifold is obtained by attaching handles in order of increasing index
• The topology of the manifold can be reconstructed from the handlebody decomposition by gluing the handles together according to the attaching maps
• The handlebody decomposition is not unique, but different decompositions are related by a sequence of handle slides and cancellations
• Morse homology provides an algebraic framework for studying the handlebody decomposition and its invariance properties

Applications in Topology

• Morse theory provides a powerful tool for studying the topology of manifolds
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
  • The Betti numbers measure the ranks of the homology groups and provide information about the number of holes in each dimension
• The Euler characteristic of a compact manifold can be computed as the alternating sum of the numbers of critical points of each index
• Morse theory can be used to prove the h-cobordism theorem, which characterizes the topology of cobordisms between manifolds
• The Morse homology theory provides a way to compute the homology groups of a manifold using the critical points of a Morse function
  • Morse homology is isomorphic to singular homology, but it provides a more geometric and computable approach
• Morse theory has applications to the study of geodesics on Riemannian manifolds and the topology of loop spaces in algebraic topology

Advanced Topics and Current Research

• The Morse-Smale complex is a cellular decomposition of a manifold determined by the gradient flow of a Morse function
  • It provides a more refined description of the topology than the handlebody decomposition
• Morse-Bott theory is a generalization of Morse theory that allows for degenerate critical points, which form submanifolds of the domain
• Floer homology is a powerful tool in symplectic geometry that can be viewed as an infinite-dimensional analogue of Morse homology
  • It has applications to the study of knots, 3-manifolds, and mirror symmetry
• Morse theory has been extended to study functions on stratified spaces and singular spaces, such as algebraic varieties
• Discrete Morse theory is a combinatorial analogue of Morse theory that studies functions on cell complexes
  • It has applications to topological data analysis and computational topology
• Current research in Morse theory includes the study of Morse-Novikov theory for circle-valued functions, the relationship between Morse theory and persistent homology, and the application of Morse theory to the study of moduli spaces in algebraic geometry