🗺️Morse Theory Unit 12 – Morse Theory and Cobordism

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 (examples: circle, torus, sphere)
• A smooth function is a function that has derivatives of all orders (examples: $f(x) = x^2$, $g(x, y) = \sin(x) + \cos(y)$)
  • Smooth functions are used to analyze the shape and structure of manifolds
• Critical points are points where the gradient of a smooth function vanishes (examples: local minima, local maxima, saddle points)
  • The type of critical point is determined by the Hessian matrix of the function at that point
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • The index characterizes the local behavior of the function near the critical point
• Morse functions are smooth functions with non-degenerate critical points (the Hessian matrix at each critical point has full rank)
  • Morse functions provide a way to decompose a manifold into simple pieces

Historical Context and Development

• Marston Morse (1892-1977) introduced the fundamental ideas of Morse theory in the 1920s and 1930s
  • Morse's work was motivated by problems in calculus of variations and dynamical systems
• Morse's key insight was to relate the topology of a manifold to the critical points of a smooth function defined on it
• In the 1940s and 1950s, René Thom and John Milnor further developed Morse theory and its applications
  • Thom introduced the concept of cobordism and studied the relationship between Morse theory and cobordism
  • Milnor used Morse theory to study the topology of manifolds and prove important results in differential topology
• Stephen Smale's work in the 1960s on generalized Morse theory and the h-cobordism theorem had a significant impact on the field
• In recent decades, Morse theory has found applications in various areas of mathematics, including symplectic geometry, gauge theory, and topological data analysis

Fundamental Principles of Morse Theory

• Morse theory relates the topology of a manifold to the critical points of a smooth function defined on it
• The main idea is to use the level sets of a Morse function to decompose the manifold into simple pieces
  • Level sets are the preimages of a function at a specific value (example: $f^{-1}(c) = \{x \in M : f(x) = c\}$)
• As the value of the function increases, the topology of the level sets changes only at critical points
• The index of a critical point determines how the topology of the level sets changes when passing through that point
  • For example, when passing through a critical point of index 0 (local minimum), a new connected component appears in the level set
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
  • Betti numbers are topological invariants that measure the number of holes in each dimension of the manifold
• The Morse-Smale complex is a cellular decomposition of the manifold based on the gradient flow of a Morse function
  • It provides a way to visualize the relationship between critical points and the topology of the manifold

Morse Functions and Critical Points

• A Morse function is a smooth function with non-degenerate critical points
  • Non-degenerate means that the Hessian matrix at each critical point has full rank (no zero eigenvalues)
• Examples of Morse functions include height functions on surfaces and distance functions from a point
• Critical points of a Morse function are classified by their index, which is the number of negative eigenvalues of the Hessian matrix
  • Index 0: local minimum
  • Index 1: saddle point with one negative eigenvalue
  • Index 2: saddle point with two negative eigenvalues (on a surface)
  • Index n: local maximum (on an n-dimensional manifold)
• The Morse lemma states that near a non-degenerate critical point, a Morse function can be locally expressed as a quadratic form
  • This local form determines the behavior of the function near 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
  • This property allows the use of Morse functions to study the topology of manifolds

Gradient Flows and Trajectories

• The gradient of a smooth function $f$ on a Riemannian manifold $M$ is a vector field $\nabla f$ that points in the direction of steepest ascent
  • The gradient depends on the choice of Riemannian metric on the manifold
• The gradient flow of a Morse function is the flow generated by the negative gradient vector field $-\nabla f$
  • Trajectories of the gradient flow are curves that follow the direction of steepest descent
• Critical points of the Morse function are equilibrium points of the gradient flow
  • The index of a critical point determines the stability of the corresponding equilibrium point
• Stable and unstable manifolds of a critical point are subsets of the manifold defined by the behavior of the gradient flow near the critical point
  • The stable manifold consists of points whose trajectories converge to the critical point as time goes to infinity
  • The unstable manifold consists of points whose trajectories converge to the critical point as time goes to negative infinity
• The Morse-Smale condition requires that stable and unstable manifolds of critical points intersect transversely
  • Transverse intersection means that the tangent spaces of the manifolds at the intersection point span the entire tangent space of the ambient manifold
• Gradient flows and trajectories provide a dynamical perspective on Morse theory and help understand the relationship between critical points and the topology of the manifold

Handlebody Decomposition

• A handlebody is a special type of manifold obtained by attaching handles to a ball
  • A handle of index $k$ is a product of a $k$-dimensional ball and an $(n-k)$-dimensional ball, where $n$ is the dimension of the manifold
• Morse theory allows for the decomposition of a manifold into a union of handlebodies, known as a handlebody decomposition
• Each critical point of a Morse function corresponds to the attachment of a handle
  • The index of the critical point determines the index of the attached handle
• Handlebody decomposition provides a way to build up a manifold from simple pieces (handles) based on the critical points of a Morse function
  • Starting from a minimum, handles are attached in order of increasing index
• The attaching map of a handle specifies how the boundary of the handle is glued to the boundary of the existing manifold
  • The attaching map is determined by the stable manifold of the corresponding critical point
• Cancellation of critical points corresponds to the cancellation of handles in the handlebody decomposition
  • A pair of critical points of consecutive indices can be cancelled if their stable and unstable manifolds intersect transversely
• Handlebody decomposition is a powerful tool for understanding the topology of manifolds and has applications in various areas, such as surgery theory and 4-manifold topology

Applications in Topology and Geometry

• Morse theory has numerous applications in topology and geometry, providing insights into the structure and properties of manifolds
• In differential topology, Morse theory is used to study the topology of smooth manifolds
  • Morse functions can be used to compute topological invariants, such as homology and cohomology groups
  • The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
• In Riemannian geometry, Morse theory is applied to the study of geodesics and the topology of Riemannian manifolds
  • The energy functional on the space of paths in a Riemannian manifold is a Morse function whose critical points correspond to geodesics
  • Morse theory can be used to prove the sphere theorem, which characterizes manifolds with positive curvature
• In symplectic geometry, Morse theory is generalized to the study of Hamiltonian systems and Floer homology
  • Floer homology is a powerful tool for studying the intersection properties of Lagrangian submanifolds in symplectic manifolds
• In topological data analysis, Morse theory is used to analyze the shape and structure of high-dimensional datasets
  • Morse functions can be used to construct simplicial complexes (such as the Morse-Smale complex) that capture the topological features of the data
• Other applications of Morse theory include the study of knots and links, the topology of complex algebraic varieties, and the analysis of dynamical systems

Cobordism Theory and Its Connection to Morse Theory

• Cobordism theory studies the relationships between manifolds of the same dimension
  • Two manifolds are cobordant if their disjoint union is the boundary of a higher-dimensional manifold (called a cobordism)
• Cobordism defines an equivalence relation on the set of manifolds, and the resulting equivalence classes form the cobordism groups
  • Cobordism groups are important algebraic invariants that capture global properties of manifolds
• Morse theory provides a natural connection between cobordism and the study of critical points of functions
• The Morse-Smale condition ensures that the cobordism between two level sets of a Morse function is well-behaved
  • The cobordism can be decomposed into a union of handles, each corresponding to a critical point of the Morse function
• The Thom-Pontryagin construction relates cobordism groups to homotopy groups of certain classifying spaces (Thom spaces)
  • This construction allows for the computation of cobordism groups using algebraic topology techniques
• The Morse-Floer homology is a generalization of Morse homology that incorporates cobordism information
  • It provides a powerful tool for studying the relationships between critical points and the topology of the underlying manifold
• Cobordism theory has applications in various areas of mathematics, including algebraic topology, differential topology, and mathematical physics
  • In physics, cobordisms are used to describe the evolution of spacetime and the behavior of quantum fields

Advanced Topics and Current Research

• Morse theory has been generalized and extended in various directions, leading to active research areas in modern mathematics
• Floer homology is a generalization of Morse homology to infinite-dimensional settings, such as the study of symplectic manifolds and gauge theory
  • Floer homology has been used to prove important results, such as the Arnold conjecture on the number of fixed points of Hamiltonian diffeomorphisms
• Morse-Bott theory is an extension of Morse theory that allows for degenerate critical points, where the Hessian matrix may have zero eigenvalues
  • Morse-Bott functions have critical submanifolds instead of isolated critical points
  • Morse-Bott theory has applications in the study of moment maps and equivariant cohomology
• Persistent homology is a technique in topological data analysis that uses ideas from Morse theory to study the multi-scale topology of datasets
  • It allows for the identification of topological features that persist across different scales and provides a way to visualize the shape of high-dimensional data
• Morse theory has been applied to the study of minimal surfaces and the Plateau problem, which seeks to find surfaces of minimal area with a given boundary
  • The Morse index of a minimal surface is related to its stability properties and the topology of the ambient manifold
• Current research in Morse theory includes the study of infinite-dimensional manifolds, the development of computational methods for Morse homology, and the application of Morse theory to problems in mathematical physics and data analysis
  • These advances continue to demonstrate the power and versatility of Morse theory as a fundamental tool in modern mathematics