🔢Algebraic Topology Unit 12 – Morse Theory

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 manifold is a topological space that locally resembles Euclidean space near each point (e.g., a circle, a sphere, or a torus)
• A smooth function is a function that has derivatives of all orders everywhere in its domain
  • Examples include polynomials, exponential functions, and trigonometric functions
• Critical points are points where the gradient of the function vanishes (i.e., where the derivative is zero)
  • Types of critical points include minima, maxima, and saddle points
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
• The Morse index theorem relates the index of a critical point to the topology of the manifold
• A Morse function is a smooth function whose critical points are non-degenerate (i.e., the Hessian matrix is non-singular at those points)

Historical Context and Development

• Morse theory was developed by American mathematician Marston Morse in the 1920s and 1930s
• Morse was motivated by the problem of understanding the topology of a manifold through the study of smooth functions on it
• The early work of Morse focused on the relationship between the number of critical points of a function and the Betti numbers of the manifold
  • Betti numbers are topological invariants that measure the connectivity of a space
• In the 1940s and 1950s, René Thom and John Milnor made significant contributions to Morse theory
  • Thom introduced the concept of cobordism, which relates Morse theory to differential topology
  • Milnor developed the theory of handle decompositions, which provides a way to construct manifolds from simple building blocks
• In recent decades, Morse theory has found applications in various areas of mathematics and physics, including symplectic geometry, gauge theory, and string theory

Fundamental Principles of Morse Theory

• The main idea of Morse theory is to study the topology of a manifold by analyzing the critical points of a smooth function defined on it
• The critical points of a Morse function provide a way to decompose the manifold into simple pieces (called handles)
• The attachment of handles is determined by the index of the critical points
  • A critical point of index $k$ corresponds to the attachment of a $k$-dimensional handle
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
  • Specifically, the Morse inequalities state that the number of critical points of index $k$ is greater than or equal to the $k$-th Betti number
• The Morse-Smale complex is a cellular decomposition of the manifold based on the gradient flow of the Morse function
  • The cells of the Morse-Smale complex are determined by the stable and unstable manifolds of the critical points
• The handle decomposition and the Morse-Smale complex provide complementary ways to understand the topology of the manifold

Morse Functions and Critical Points

• A Morse function is a smooth function whose critical points are non-degenerate
• The Hessian matrix of a Morse function at a critical point has non-zero determinant
  • The Hessian matrix is the matrix of second partial derivatives of the function
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • A minimum has index 0, a saddle point has index 1, and a maximum has index 2 (in the case of a surface)
• The Morse lemma states that near a non-degenerate critical point, a Morse function can be written in a standard quadratic form
  • The quadratic form is determined by the index of the critical point
• Morse functions are dense in the space of smooth functions, meaning that any smooth function can be approximated by a Morse function
• The critical points of a Morse function are isolated, and their indices satisfy the Morse inequalities
  • The Morse inequalities provide a lower bound for the number of critical points of each index

Gradient Flow and Vector Fields

• The gradient of a smooth function is a vector field that points in the direction of steepest ascent
  • The gradient is defined as the vector of partial derivatives of the function
• The gradient flow of a Morse function is a dynamical system that describes the evolution of points under the gradient vector field
  • The gradient flow is defined by the ordinary differential equation $\dot{x} = \nabla f(x)$, where $f$ is the Morse function
• The stable manifold of a critical point is the set of points that flow to the critical point under the gradient flow
  • The stable manifold has dimension equal to the index of the critical point
• The unstable manifold of a critical point is the set of points that flow away from the critical point under the gradient flow
  • The unstable manifold has dimension equal to the codimension of the index of the critical point
• The stable and unstable manifolds of critical points intersect transversely, forming the Morse-Smale complex
• The gradient flow provides a way to understand the attachment of handles in the handle decomposition of the manifold

Homotopy and Homology in Morse Theory

• Morse theory provides a connection between the homotopy type of a manifold and the homology of the Morse-Smale complex
• The homotopy type of a manifold is determined by its critical points and their indices
  • Specifically, the manifold is homotopy equivalent to a CW complex built from the critical points, with cells of dimension equal to the index
• The homology of the Morse-Smale complex is isomorphic to the homology of the manifold
  • The boundary operator in the Morse-Smale complex is determined by the gradient flow between critical points
• The Morse homology theorem states that the Morse homology (defined using the Morse-Smale complex) is isomorphic to the singular homology of the manifold
• The Morse inequalities can be interpreted as a consequence of the relationship between Morse homology and singular homology
  • The Morse inequalities provide a lower bound for the Betti numbers of the manifold
• The Morse-Witten complex is a chain complex that computes the Morse homology using the gradient flow between critical points

Applications in Algebraic Topology

• Morse theory has numerous applications in algebraic topology, providing a bridge between the analytic and topological aspects of the subject
• Morse theory can be used to compute the homology and cohomology of manifolds
  • The Morse-Smale complex provides a cellular decomposition that can be used to define the chain complexes
• Morse theory is a key tool in the study of cobordism, which relates manifolds of different dimensions
  • Two manifolds are cobordant if their disjoint union is the boundary of a higher-dimensional manifold
• Morse theory plays a role in the h-cobordism theorem, which characterizes cobordisms between simply-connected manifolds
  • The h-cobordism theorem states that a cobordism between simply-connected manifolds is trivial if and only if it admits a Morse function with no critical points
• Morse theory has applications in the study of exotic spheres, which are spheres that are homeomorphic but not diffeomorphic to the standard sphere
  • Milnor used Morse theory to construct exotic 7-spheres, demonstrating the existence of smooth structures on spheres
• Morse theory is also used in the study of the topology of complex algebraic varieties, through the theory of Lefschetz pencils and fibrations

Advanced Topics and Current Research

• Morse theory has been generalized in various directions, leading to the development of new techniques and applications
• The Morse-Bott theory extends Morse theory to the case where the critical points form submanifolds, rather than being isolated
  • Morse-Bott functions have degenerate critical points, but the Hessian matrix is non-degenerate in the normal directions
• Floer homology is a variant of Morse homology that is defined for infinite-dimensional manifolds, such as the loop space of a manifold
  • Floer homology has important applications in symplectic geometry and low-dimensional topology
• Morse theory has been applied to the study of minimal surfaces and harmonic maps between Riemannian manifolds
  • The Morse index of a minimal surface or harmonic map is related to its stability properties
• Morse theory has connections to physics, particularly in the areas of quantum field theory and string theory
  • The Morse theory of loop spaces is related to the path integral formulation of quantum mechanics
• Current research in Morse theory includes the study of Morse-Smale complexes on infinite-dimensional manifolds, the relationship between Morse theory and persistent homology, and the application of Morse theory to data analysis and machine learning.