11.3 Morse theory

Morse theory is a powerful tool for understanding the topology of smooth manifolds. By analyzing critical points of real-valued functions, it reveals crucial information about a manifold's structure and properties.

This approach connects geometry and topology, allowing us to decompose manifolds into simpler pieces. Morse theory's applications range from proving topological results to solving problems in physics and dynamical systems.

Morse functions

• Morse functions are real-valued functions on a smooth manifold that satisfy certain non-degeneracy conditions at their critical points
• They play a crucial role in studying the topology of smooth manifolds by analyzing the behavior of the function near its critical points
• Morse functions provide a way to decompose a manifold into simple pieces, allowing for a deeper understanding of its structure

Critical points of Morse functions

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• Critical points of a Morse function are points where the gradient of the function vanishes
• At a critical point, the Hessian matrix (matrix of second partial derivatives) of the Morse function is non-degenerate
• The index of a critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • Index 0 critical points correspond to local minima
  • Index n critical points (where n is the dimension of the manifold) correspond to local maxima
  • Index k critical points (0 < k < n) correspond to saddle points

Non-degenerate critical points

• A critical point is non-degenerate if the Hessian matrix at that point is invertible
• Non-degenerate critical points are isolated, meaning there exists a neighborhood around the point containing no other critical points
• The behavior of a Morse function near a non-degenerate critical point is completely determined by its index
  • For example, near an index 1 critical point, the function has a saddle shape

Morse lemma

• The Morse lemma states that in a neighborhood of a non-degenerate critical point, a Morse function can be expressed as a quadratic form in some local coordinates

• Specifically, if p is a critical point of index k, then there exist local coordinates $(x_1, \ldots, x_n)$ centered at p such that the Morse function f can be written as: $f(x) = f(p) - x_1^2 - \ldots - x_k^2 + x_{k+1}^2 + \ldots + x_n^2$

• The Morse lemma provides a standard form for Morse functions near their critical points, which is essential for studying the local behavior of the function

Morse-Smale functions

• Morse-Smale functions are a special class of Morse functions that satisfy an additional transversality condition
• They are named after Marston Morse and Stephen Smale, who made significant contributions to the theory
• Morse-Smale functions have well-behaved gradient flows, which makes them particularly useful for studying the dynamics and topology of manifolds

Stable and unstable manifolds

• For each critical point p of a Morse-Smale function, there are two associated submanifolds: the stable manifold $W^s(p)$ and the unstable manifold $W^u(p)$
• The stable manifold $W^s(p)$ consists of all points that flow to p under the gradient flow of the Morse-Smale function as time goes to infinity
• The unstable manifold $W^u(p)$ consists of all points that flow to p under the gradient flow as time goes to negative infinity
• The dimension of $W^s(p)$ is equal to the index of p, while the dimension of $W^u(p)$ is equal to the codimension of p (dimension of the manifold minus the index)

Transversality of stable and unstable manifolds

• In a Morse-Smale function, the stable and unstable manifolds of any two critical points intersect transversally
• Transversality means that the tangent spaces of the stable and unstable manifolds at any point of intersection span the entire tangent space of the manifold at that point
• The transversality condition ensures that the intersections between stable and unstable manifolds are well-behaved and have the expected dimension

Morse-Smale gradient flows

• The gradient flow of a Morse-Smale function is a dynamical system on the manifold, where each point moves in the direction of the negative gradient of the function
• The transversality condition implies that the gradient flow is structurally stable, meaning that small perturbations of the function do not change the qualitative behavior of the flow
• Morse-Smale gradient flows have a finite number of critical points and a finite number of connecting orbits between them, forming a directed graph called the Morse-Smale complex
  • The vertices of the Morse-Smale complex are the critical points
  • The edges correspond to the connecting orbits between critical points

Morse homology

• Morse homology is a homology theory constructed from the critical points and gradient flow lines of a Morse function
• It provides a way to compute the homology groups of a manifold using the information encoded in a Morse function
• Morse homology is isomorphic to singular homology, which is a classical homology theory defined using singular simplices

Morse complex

• The Morse complex is a chain complex associated with a Morse function on a manifold
• The chain groups of the Morse complex are generated by the critical points of the Morse function, graded by their index
• The boundary operator of the Morse complex counts the gradient flow lines between critical points of consecutive indices
  • Specifically, the boundary of a critical point p of index k is the sum of all critical points q of index k-1, weighted by the number of gradient flow lines (modulo 2) from p to q

Boundary operator in Morse homology

• The boundary operator $\partial$ in Morse homology satisfies the property that $\partial \circ \partial = 0$, making the Morse complex a well-defined chain complex
• The proof of this property relies on the analysis of the compactified moduli space of gradient flow lines and the cancellation of boundary terms
• The homology of the Morse complex, obtained by taking the quotient of the kernel of $\partial$ by its image, is called Morse homology

Morse homology vs singular homology

• The Morse homology groups are isomorphic to the singular homology groups of the manifold
• This isomorphism is established by constructing a chain map between the Morse complex and the singular chain complex, and showing that it induces an isomorphism on homology
• The Morse homology approach provides a more geometric and computational way to calculate homology groups, as it relies on the critical points and gradient flow lines of a Morse function

Morse inequalities

• Morse inequalities are a set of relations between the number of critical points of a Morse function and the Betti numbers of the manifold
• They provide a powerful tool for studying the topology of manifolds using Morse theory
• Morse inequalities come in two forms: weak and strong

Weak Morse inequalities

• The weak Morse inequalities state that for a Morse function f on a closed manifold M, the number $c_k$ of critical points of index k satisfies: $c_k \geq b_k(M)$

where $b_k(M)$ is the k-th Betti number of M (the rank of the k-th homology group)

• In other words, the number of critical points of index k is always greater than or equal to the k-th Betti number
• The weak Morse inequalities hold for any Morse function on the manifold

Strong Morse inequalities

• The strong Morse inequalities provide a more refined relationship between the critical points and the Betti numbers

• They state that for a Morse function f on a closed manifold M, the following inequality holds for all k: $\sum_{i=0}^k (-1)^{k-i} c_i \geq \sum_{i=0}^k (-1)^{k-i} b_i(M)$

• The strong Morse inequalities imply the weak Morse inequalities, but not vice versa

• The equality in the strong Morse inequalities holds if and only if the Morse function is perfect, meaning that the Morse complex has no non-trivial boundary operators

Morse inequalities and Betti numbers

• Morse inequalities provide lower bounds for the Betti numbers of a manifold in terms of the critical points of a Morse function
• They can be used to prove the existence of certain topological features, such as non-trivial homology groups or the presence of handles of a certain index
• In some cases, Morse inequalities can determine the Betti numbers of a manifold completely, especially when the Morse function is perfect
  • For example, if a closed manifold admits a perfect Morse function with exactly two critical points (a minimum and a maximum), then the manifold must be a sphere

Morse theory and handle decompositions

• Morse theory is closely related to the concept of handle decompositions of manifolds
• A handle decomposition is a way to build a manifold by attaching handles of various dimensions along their boundaries
• Morse functions provide a natural way to construct handle decompositions, with each critical point corresponding to the attachment of a handle

Critical points and handles

• Each critical point of index k in a Morse function corresponds to the attachment of a k-handle to the manifold
• A k-handle is a product of a k-dimensional disk and an (n-k)-dimensional disk, where n is the dimension of the manifold
• The attachment of a k-handle is performed along the boundary of the k-dimensional disk, which is an (k-1)-dimensional sphere
• The stable manifold of a critical point of index k corresponds to the core disk of the associated k-handle, while the unstable manifold corresponds to the co-core disk

Existence of Morse functions

• A fundamental result in Morse theory states that every smooth manifold admits a Morse function
• Moreover, any two Morse functions on a manifold are related by a sequence of handle slides and birth-death bifurcations
• This result implies that the handle decomposition of a manifold is unique up to these operations, providing a powerful tool for studying the topology of manifolds

Morse theory and surgery

• Morse theory is closely related to the theory of surgery on manifolds
• Surgery is a technique for modifying a manifold by cutting out a submanifold and replacing it with another submanifold with the same boundary
• The process of attaching handles in a handle decomposition can be interpreted as performing surgery on the manifold
• Morse functions provide a systematic way to describe and analyze the surgery process, by encoding the necessary information in the critical points and their indices

Applications of Morse theory

• Morse theory has numerous applications in various areas of mathematics and physics
• It provides a powerful tool for studying the topology of manifolds, as well as the behavior of dynamical systems and variational problems
• Some notable applications of Morse theory include:

Morse theory and topology of manifolds

• Morse theory can be used to prove fundamental results in the topology of manifolds, such as the Poincaré conjecture in dimensions greater than 4
• By studying the critical points of a Morse function and their indices, one can obtain information about the homology groups, homotopy groups, and other topological invariants of the manifold
• Morse theory also provides a way to construct and classify manifolds, by specifying the number and type of critical points of a Morse function

Morse theory in symplectic geometry

• Morse theory has important applications in symplectic geometry, which studies symplectic manifolds and their properties
• In this context, Morse functions are replaced by Morse-Bott functions, which allow for critical submanifolds instead of isolated critical points
• The study of Morse-Bott functions on symplectic manifolds leads to the development of Floer homology, which is a powerful tool for understanding the intersection properties of Lagrangian submanifolds and the dynamics of Hamiltonian systems

Morse theory in mathematical physics

• Morse theory has significant applications in various areas of mathematical physics, such as quantum field theory, string theory, and general relativity
• In quantum field theory, Morse theory is used to study the topology of the configuration space of fields and to define topological invariants, such as the Witten index
• In string theory, Morse theory is applied to the study of the landscape of vacua and the properties of the moduli space of string compactifications
• In general relativity, Morse theory is used to analyze the topology of the space of solutions to the Einstein equations and to study the properties of gravitational instantons