🗺️Morse Theory Unit 10 – The Morse Inequalities

Key Concepts and Definitions

• Morse function $f: M \to \mathbb{R}$ smooth function on a smooth manifold $M$ with non-degenerate critical points
• Critical point $p \in M$ point where the gradient of the Morse function vanishes $\nabla f(p) = 0$
  • Non-degenerate critical point has a non-singular Hessian matrix at $p$
• Index of a critical point number of negative eigenvalues of the Hessian matrix at that point
• $M_a$ sublevel set of $M$ defined as $\{x \in M : f(x) \leq a\}$ for some $a \in \mathbb{R}$
• $\beta_k(M_a)$ $k$-th Betti number of the sublevel set $M_a$, representing the rank of the $k$-th homology group
• $C_k(f)$ number of critical points of $f$ with index $k$
• Weak Morse inequalities relate the Betti numbers and the number of critical points $\beta_k(M) \leq C_k(f)$

Historical Context and Development

• Marston Morse introduced Morse theory in the 1920s as a tool to study the topology of smooth manifolds
• Morse's initial work focused on understanding the relationship between the critical points of a function and the topology of the underlying manifold
• In the 1930s, Morse established the strong and weak Morse inequalities, providing a quantitative link between critical points and homology
• Contributions from René Thom, Stephen Smale, and John Milnor in the 1950s and 1960s further developed Morse theory and its applications
  • Thom's work on cobordism and catastrophe theory
  • Smale's proof of the generalized Poincaré conjecture for dimensions $n \geq 5$ using Morse theory
• Morse theory has since become a fundamental tool in differential topology and geometry, with applications in various fields of mathematics and physics

Statement of the Morse Inequalities

• Let $M$ be a compact smooth manifold and $f: M \to \mathbb{R}$ a Morse function
• Weak Morse Inequalities: For each $k$, the $k$-th Betti number is bounded above by the number of critical points of index $k$
  • $\beta_k(M) \leq C_k(f)$
• Strong Morse Inequalities: Alternating sums of Betti numbers and critical points satisfy
  • $\sum_{i=0}^k (-1)^{k-i} \beta_i(M) \leq \sum_{i=0}^k (-1)^{k-i} C_i(f)$ for all $k$
  • Equality holds when $k = \dim(M)$
• Morse Lacunary Principle: If $M$ has no critical points of index $k$, then $\beta_k(M) = \beta_{k-1}(M)$
• Morse inequalities provide a lower bound for the number of critical points based on the homology of the manifold

Proof Techniques and Strategies

• Prove the weak Morse inequalities using induction on sublevel sets and the Morse Lemma
  • Show that passing a non-degenerate critical point of index $k$ changes the homology by at most rank 1 in dimension $k$
• Establish the strong Morse inequalities by considering the Euler characteristic and Poincaré polynomial
  • Express the Euler characteristic as alternating sums of Betti numbers and critical points
  • Compare coefficients of the Poincaré polynomials for the homology and the Morse function
• Utilize the Morse-Smale complex, a cellular decomposition of the manifold based on the gradient flow of the Morse function
  • Each cell corresponds to a critical point, with the dimension equal to the index
• Apply the Conley index theory to study isolated invariant sets and their homology
• Use the Witten deformation technique to relate Morse theory to supersymmetric quantum mechanics

Applications in Topology and Geometry

• Prove the existence of a minimal number of critical points on compact manifolds
  • Sphere $S^n$ has at least 2 critical points (minimum and maximum)
  • Torus $T^n$ has at least $2^n$ critical points
• Classify surfaces by studying Morse functions and their critical points
• Compute homology groups and Betti numbers using Morse inequalities and Morse homology
• Study the topology of sublevel sets and level sets of Morse functions
  • Reeb graph encodes the evolution of level sets
• Investigate the topology of solution spaces in variational problems and PDEs
• Apply Morse theory to Riemannian and Finsler geometry to study geodesics and curvature

Connections to Other Mathematical Theories

• Floer homology: Infinite-dimensional analog of Morse homology for studying symplectic geometry and low-dimensional topology
  • Floer complexes generated by critical points of the action functional on the loop space
• Gauge theory: Morse-Bott functions used to study the topology of moduli spaces of connections
• Topological data analysis: Morse-Smale complexes and Reeb graphs used for data visualization and feature extraction
• Symplectic topology: Morse theory on the loop space of a symplectic manifold
• Algebraic topology: Morse homology as an alternative approach to singular homology
  • Morse inequalities provide a link between critical points and the ranks of homology groups
• Differential equations: Morse theory used to study the qualitative behavior of solutions and bifurcations
• Quantum field theory: Morse theory appears in the context of supersymmetric quantum mechanics and topological quantum field theories

Examples and Problem-Solving

• Height function on the torus $T^2$: 4 critical points (1 minimum, 2 saddles, 1 maximum)
  • Betti numbers: $\beta_0 = 1, \beta_1 = 2, \beta_2 = 1$
  • Weak Morse inequalities: $1 \leq 1, 2 \leq 2, 1 \leq 1$
• Morse function on the real projective plane $\mathbb{RP}^2$: 3 critical points (1 minimum, 1 saddle, 1 maximum)
  • Betti numbers: $\beta_0 = 1, \beta_1 = 0, \beta_2 = 1$
  • Strong Morse inequalities: $1 \leq 1, 1 \leq 2, 0 \leq 1$
• Height function on the 2-sphere $S^2$: 2 critical points (1 minimum, 1 maximum)
  • Betti numbers: $\beta_0 = 1, \beta_1 = 0, \beta_2 = 1$
  • Morse Lacunary Principle: $\beta_1 = \beta_0 = 1$ (no critical points of index 1)
• Studying the topology of energy landscapes in physical systems using Morse theory
• Analyzing the critical points of the distance function on a Riemannian manifold

Advanced Topics and Current Research

• Morse-Bott theory: Generalization of Morse theory to functions with degenerate critical submanifolds
  • Morse-Bott inequalities relating Betti numbers to the indices of critical submanifolds
• Equivariant Morse theory: Study of Morse functions invariant under a group action
  • Equivariant homology and cohomology, equivariant Morse inequalities
• Infinite-dimensional Morse theory: Extension of Morse theory to infinite-dimensional manifolds and Hilbert spaces
  • Applications in variational analysis, nonlinear PDEs, and Hamiltonian systems
• Discrete Morse theory: Combinatorial analog of Morse theory for cell complexes
  • Forman's discrete Morse inequalities, discrete Morse functions, and gradient vector fields
• Stochastic Morse theory: Morse theory for stochastic differential equations and random dynamical systems
• Morse-Conley-Floer theory: Unification of Morse theory, Conley index theory, and Floer homology
  • Studying dynamical systems, symplectic geometry, and low-dimensional topology
• Persistent homology: Combining Morse theory with algebraic topology to study the persistence of topological features across scales