🗺️Morse Theory Unit 7 – Morse Functions and Topology

Foundations of Morse Theory

• Morse theory studies the relationship between the topology of a manifold and the critical points of a smooth function defined on it
• Developed by American mathematician Marston Morse in the early 20th century
• Combines techniques from differential topology, dynamical systems, and algebraic topology
• Provides a powerful tool for understanding the global structure of a manifold through local information about critical points
• Has applications in various fields such as physics, biology, and computer graphics
  • In physics, used to study the topology of energy landscapes and phase transitions
  • In biology, helps analyze the shape and connectivity of biomolecules
  • In computer graphics, used for surface reconstruction and mesh simplification

Key Concepts in Topology

• Topology is the study of properties that are preserved under continuous deformations (stretching, twisting, bending) but not tearing or gluing
• Manifolds are topological spaces that locally resemble Euclidean space and form the main objects of study in Morse theory
  • Examples include spheres, tori, and projective spaces
• Homeomorphisms are continuous bijections with continuous inverses that define topological equivalence between spaces
• Homotopy is a continuous deformation between two continuous functions or paths
  • Two spaces are homotopy equivalent if there exist continuous maps between them that are homotopy inverses
• Homology is an algebraic invariant that captures the "holes" in a topological space
  • Computed using chain complexes and boundary operators
  • Betti numbers are the ranks of homology groups and provide a coarse measure of the topology

Morse Functions: Definition and Properties

• A Morse function is a smooth real-valued function $f: M \to \mathbb{R}$ on a manifold $M$ whose critical points are non-degenerate
• Critical points are points where the gradient $\nabla f$ vanishes
  • Correspond to local extrema (minima, maxima) and saddle points
• Non-degenerate critical points have a non-singular Hessian matrix (matrix of second partial derivatives)
• Morse functions are generic in the sense that they form an open dense subset of the space of smooth functions
• The Morse index of a critical point is the number of negative eigenvalues of the Hessian matrix
  • Determines the local behavior of the function near the critical point
• Morse functions satisfy the Morse inequalities relating the critical points to the Betti numbers of the manifold

Critical Points and Their Classification

• Critical points of a Morse function are classified by their Morse index
  • Index 0: local minimum
  • Index 1: saddle point with one negative direction
  • Index 2: saddle point with two negative directions
  • ...
  • Index $n$: local maximum (for an $n$-dimensional manifold)
• The Morse lemma states that near a non-degenerate critical point, the function can be written in a standard quadratic form
• The number of critical points of each index is related to the Betti numbers of the manifold via the Morse inequalities
  • $m_k \geq b_k$ where $m_k$ is the number of critical points of index $k$ and $b_k$ is the $k$-th Betti number
• Morse functions on closed manifolds must have at least two critical points (a minimum and a maximum)
• The Euler characteristic of the manifold can be expressed as the alternating sum of critical points: $\chi(M) = \sum_{k=0}^n (-1)^k m_k$

Gradient Vector Fields and Flow Lines

• The gradient vector field of a Morse function $f$ is the vector field $\nabla f$ that points in the direction of steepest ascent
• Integral curves of the gradient vector field are called flow lines or gradient trajectories
  • They are curves $\gamma(t)$ that satisfy $\gamma'(t) = \nabla f(\gamma(t))$
• Flow lines originate and terminate at critical points
  • Stable manifold of a critical point: set of points whose flow lines converge to the critical point as $t \to \infty$
  • Unstable manifold: set of points whose flow lines converge to the critical point as $t \to -\infty$
• The stable and unstable manifolds of critical points form a cell decomposition of the manifold called the Morse complex
• Morse homology is defined using the Morse complex and is isomorphic to the singular homology of the manifold

The Morse Lemma and Local Behavior

• The Morse lemma provides a local normal form for a Morse function near a non-degenerate critical point
• States that in a suitable coordinate system $(x_1, \ldots, x_n)$ centered at the critical point, the function can be written as $f(x) = f(p) - x_1^2 - \ldots - x_k^2 + x_{k+1}^2 + \ldots + x_n^2$
  • $k$ is the Morse index of the critical point
• Implies that the level sets of the function near a critical point are quadric hypersurfaces
  • For index 0 (minimum): ellipsoids
  • For index 1 (saddle): hyperboloids of one sheet
  • For index 2 (saddle): hyperboloids of two sheets
  • For index $n$ (maximum): ellipsoids
• The Morse lemma is a key tool in understanding the local topology near critical points and proving the Morse inequalities

Homotopy and Homology in Morse Theory

• Morse theory relates the critical points of a Morse function to the homotopy and homology of the manifold
• The sublevel sets $M_a = f^{-1}((-\infty, a])$ of a Morse function $f$ provide a filtration of the manifold
  • As $a$ increases, the topology of $M_a$ changes only when $a$ passes through a critical value
  • The change in topology is determined by the index of the critical point
• Attaching a $k$-cell (a $k$-dimensional disk) to $M_a$ at a critical point of index $k$ corresponds to the birth of a $k$-dimensional homology class
• The Morse inequalities relate the critical points to the Betti numbers: $m_k \geq b_k$
  • Equality holds if the Morse function is perfect (Morse complex is a CW complex)
• The Morse homology complex is constructed using the critical points as generators and the gradient flow lines as boundary operators
  • Isomorphic to the singular homology of the manifold
• The Morse-Smale complex is a finer cellular decomposition that captures the dynamics of the gradient flow

Applications in Mathematics and Beyond

• Morse theory has numerous applications in various branches of mathematics and other fields
• In differential topology, used to study the topology of manifolds and prove existence results
  • Proves the h-cobordism theorem and the high-dimensional Poincaré conjecture
• In algebraic topology, provides a way to compute homology and homotopy groups
  • Morse homology is a powerful tool for studying the topology of loop spaces and classifying spaces
• In symplectic geometry, Morse-Bott theory generalizes Morse theory to study the topology of symplectic manifolds and Hamiltonian systems
• In mathematical physics, used to study the topology of energy landscapes and phase transitions
  • Morse theory on infinite-dimensional manifolds (Floer theory) has applications in gauge theory and string theory
• In computer graphics and visualization, Morse theory is used for surface reconstruction, mesh simplification, and topological data analysis
  • Reeb graphs and Morse-Smale complexes capture the topological structure of scalar fields and shapes