🗺️Morse Theory Unit 15 – Morse Theory and Floer Homology

Key Concepts and Definitions

• Morse theory studies the relationship between the topology of a manifold and the critical points of a real-valued function defined on it
• A smooth manifold is a topological space that locally resembles Euclidean space near each point
  • Includes examples such as spheres, tori, and projective spaces
• A Morse function is a smooth real-valued function with non-degenerate critical points
  • Non-degenerate critical points have a non-singular Hessian matrix
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
• The Morse inequalities relate the number of critical points of each index to the Betti numbers of the manifold
• The Morse complex is a chain complex constructed from the critical points of a Morse function
  • Its homology is isomorphic to the singular homology of the manifold

Historical Context and Development

• Morse theory originated in the work of Marston Morse in the 1920s and 1930s
  • Morse introduced the concept of a Morse function and proved the Morse inequalities
• In the 1950s and 1960s, René Thom and Stephen Smale further developed Morse theory
  • Thom introduced the concept of cobordism and studied the topology of singularities
  • Smale proved the generalized Poincaré conjecture in higher dimensions using Morse theory
• In the 1980s, Edward Witten introduced a new perspective on Morse theory using supersymmetry and quantum field theory
  • This led to the development of Floer homology and other variants of Morse theory
• Recent developments include the study of Morse-Bott functions, which allow for degenerate critical points
  • Morse-Bott theory has applications in symplectic geometry and gauge theory

Morse Functions and Critical Points

• A critical point of a smooth function $f: M \to \mathbb{R}$ is a point where the differential $df$ vanishes
• The Hessian matrix of $f$ at a critical point $p$ is the matrix of second partial derivatives $(\partial^2 f/\partial x_i \partial x_j)(p)$
  • A critical point is non-degenerate if the Hessian matrix is non-singular
• The Morse lemma states that near a non-degenerate critical point, a Morse function can be expressed in a standard quadratic form
  • The number of negative squares in the quadratic form is the index of the critical point
• Morse functions are dense in the space of smooth functions, meaning any smooth function can be approximated by a Morse function
• The critical points of a Morse function determine a decomposition of the manifold into handles
  • The index of a critical point determines the dimension of the handle

The Gradient Flow and Morse-Smale Condition

• The gradient of a smooth function $f: M \to \mathbb{R}$ is the vector field $\nabla f$ defined by $df(v) = \langle \nabla f, v \rangle$ for all tangent vectors $v$
• The gradient flow of $f$ is the flow generated by the negative gradient $-\nabla f$
  • Integral curves of the gradient flow are paths of steepest descent for $f$
• A Morse function satisfies the Morse-Smale condition if its gradient flow is transverse to the unstable manifolds of its critical points
  • This means that the stable and unstable manifolds of distinct critical points intersect transversely
• The stable manifold of a critical point $p$ is the set of points that flow to $p$ under the gradient flow as time goes to infinity
  • The unstable manifold is the set of points that flow to $p$ as time goes to negative infinity
• The Morse-Smale condition implies that the gradient flow defines a cellular decomposition of the manifold
  • The cells are the unstable manifolds of the critical points

Morse Homology: Construction and Properties

• The Morse complex of a Morse function $f$ is the chain complex generated by the critical points of $f$
  • The differential counts gradient flow lines between critical points of adjacent indices
• The Morse homology of $f$ is the homology of the Morse complex
  • It is isomorphic to the singular homology of the manifold
• The Morse inequalities relate the Betti numbers of the manifold to the numbers of critical points of each index
  • They provide lower bounds for the Betti numbers in terms of the critical points
• The Morse complex can be defined with integer coefficients, leading to additional torsion information
• Morse homology is independent of the choice of Morse function and Riemannian metric
  • Different choices lead to chain homotopy equivalent complexes
• The Morse complex can be localized to open subsets of the manifold, leading to relative and local Morse homology

Introduction to Floer Homology

• Floer homology is an infinite-dimensional analog of Morse homology that arises in symplectic geometry
  • It is defined using the Hamiltonian flow of a symplectic manifold
• The Floer complex is generated by the fixed points of a Hamiltonian diffeomorphism
  • The differential counts pseudoholomorphic cylinders connecting the fixed points
• Floer homology is invariant under Hamiltonian isotopies of the symplectic manifold
  • It provides a powerful tool for studying symplectic topology
• Different versions of Floer homology include Hamiltonian Floer homology, Lagrangian Floer homology, and Heegaard Floer homology
  • Each version has its own geometric setup and applications
• The Arnold conjecture, which relates the number of fixed points of a Hamiltonian diffeomorphism to the topology of the manifold, can be proved using Floer homology
  • This is a major application of Floer theory in symplectic topology

Applications in Topology and Geometry

• Morse theory provides a powerful tool for studying the topology of manifolds
  • It can be used to compute homology groups, homotopy groups, and other topological invariants
• The handle decomposition of a manifold determined by a Morse function can be used to construct explicit cell structures
  • This is useful for computing cup products, intersection forms, and other cohomological operations
• Morse theory has applications in Riemannian geometry, where it relates the curvature of a manifold to its topology
  • The Morse index theorem relates the index of a geodesic to the conjugate points along it
• In complex geometry, Morse theory can be used to study the topology of complex manifolds and submanifolds
  • The complex Morse inequality relates the Hodge numbers of a complex manifold to its Morse indices
• Floer theory has applications in low-dimensional topology, where it can be used to define invariants of knots and 3-manifolds
  • Heegaard Floer homology is a powerful tool for studying the topology of 3-manifolds

Advanced Topics and Current Research

• Morse-Bott theory is a generalization of Morse theory that allows for degenerate critical points
  • It has applications in gauge theory and the study of moduli spaces
• Equivariant Morse theory studies Morse functions that are invariant under the action of a group
  • It has applications in the study of symmetric spaces and homogeneous manifolds
• Morse theory can be extended to infinite-dimensional manifolds, such as the loop space of a manifold
  • This leads to the study of Morse-Novikov theory and Floer homotopy theory
• The Fukaya category is an A-infinity category that can be defined using Lagrangian Floer homology
  • It has applications in mirror symmetry and the study of symplectic manifolds
• Spectral sequences are a powerful algebraic tool that can be used to compute Morse and Floer homology
  • The Leray-Serre spectral sequence relates the homology of a fiber bundle to its base and fiber
• Current research in Morse theory includes the study of Morse-Smale-Witten complexes, which combine Morse theory with the Witten deformation
  • These complexes have applications in supersymmetric quantum mechanics and string topology