🔁Elementary Differential Topology Unit 8 – Sard's Theorem & Transversality

Key Concepts and Definitions

• Smooth functions map between smooth manifolds preserving their differential structure
• Critical points where the differential of a smooth function is not surjective
• Regular values outputs of a smooth function that are not critical values
• Sard's theorem states that the set of critical values of a smooth function has measure zero
• Transversality two submanifolds intersect transversely if their tangent spaces at the point of intersection together generate the tangent space of the ambient manifold
  • Transverse intersections are generally "stable" under small perturbations
• Jet spaces capture higher-order derivative information of smooth functions
• Whitney topologies allow for the study of generic properties of smooth functions

Historical Context and Importance

• Arthur Sard, an American mathematician, proved his eponymous theorem in 1942
• Sard's theorem is a fundamental result in differential topology and has far-reaching consequences
• Transversality, a concept closely related to Sard's theorem, was developed by René Thom in the 1950s
• These concepts played a crucial role in the development of singularity theory and catastrophe theory
• Sard's theorem and transversality have applications in various fields:
  • Morse theory studying the topology of manifolds using critical points of smooth functions
  • Control theory analyzing the controllability and observability of dynamical systems
  • Optimization understanding the structure of critical points in optimization problems
• Understanding these concepts is essential for studying the generic behavior of smooth functions and mappings

Sard's Theorem Explained

• Consider a smooth function $f: M \to N$ between smooth manifolds $M$ and $N$
• A point $p \in M$ is a critical point of $f$ if the differential $df_p: T_pM \to T_{f(p)}N$ is not surjective
• A critical value is the image $f(p)$ of a critical point $p$, while a regular value is not the image of any critical point
• Sard's theorem states that the set of critical values of $f$ has Lebesgue measure zero in $N$
  • In other words, "most" points in the codomain are regular values
• The proof of Sard's theorem relies on the Morse-Sard lemma, which bounds the size of the set of critical points
• Sard's theorem has several generalizations, such as the Morse-Sard theorem for $C^r$ functions and the Parametric Sard theorem

Understanding Transversality

• Two submanifolds $M$ and $N$ of a smooth manifold $W$ intersect transversely at a point $p$ if $T_pM + T_pN = T_pW$
  • In other words, their tangent spaces at $p$ span the tangent space of the ambient manifold
• Transversality is a generic property: for "most" pairs of submanifolds, their intersection is transverse
• The Transversality theorem states that for a smooth map $f: M \to N$ and a submanifold $W \subset N$, the set of maps transverse to $W$ is dense in the space of all smooth maps with the Whitney $C^\infty$ topology
• Transverse intersections are "stable" under small perturbations of the submanifolds or the map
  • This stability is crucial in many applications, such as in the study of dynamical systems
• Transversality can be used to prove the existence of certain geometric objects, such as the existence of a Morse function on any smooth manifold

Applications in Differential Topology

• Sard's theorem is used to prove the Whitney Embedding Theorem, which states that any smooth manifold can be smoothly embedded into a Euclidean space of sufficiently high dimension
• In Morse theory, Sard's theorem guarantees the existence of regular values for a smooth function, which is essential for studying the topology of the manifold using level sets
• Transversality is used in the study of vector bundles and characteristic classes
  • For example, the Thom transversality theorem relates the Thom class of a vector bundle to the Euler class
• In the study of singularities, Sard's theorem and transversality are used to understand the generic types of singularities that can occur in a family of functions
• Transversality is also used in the definition and study of Lefschetz pencils, which are important tools in 4-dimensional topology

Common Misconceptions and Pitfalls

• Sard's theorem does not imply that the set of critical points has measure zero, only the set of critical values
• The converse of Sard's theorem is false: a function can have a measure zero set of critical values without being smooth
• Transversality is a generic property, but it does not hold for all pairs of submanifolds or maps
  • It is essential to verify transversality in specific cases
• Transversality is a sufficient condition for the stability of intersections, but it is not always necessary
• The Whitney $C^\infty$ topology is not the same as the uniform $C^\infty$ topology, and the choice of topology can affect the genericity of transversality
• Sard's theorem and transversality are not directly applicable to continuous or $C^k$ functions without additional assumptions

Problem-Solving Techniques

• When applying Sard's theorem, first identify the critical points of the function and then analyze the set of critical values
• To prove that a property holds for "most" functions or maps, consider using transversality and density arguments in the appropriate function space
• When studying the intersection of submanifolds, always check for transversality
  • If the intersection is not transverse, consider perturbing the submanifolds or the map to achieve transversality
• In applications, it is often useful to work with parametrized families of functions or maps and apply the Parametric Sard theorem or the Transversality theorem
• When dealing with singular spaces or non-smooth functions, consider approximating them with smooth objects and applying Sard's theorem or transversality to the approximations

Connections to Other Topics

• Sard's theorem and transversality are closely related to the concept of genericity in dynamical systems and singularity theory
• In Morse theory, Sard's theorem is used in conjunction with the Morse Lemma to study the topology of manifolds using critical points of smooth functions
• Transversality plays a crucial role in the study of intersection theory in algebraic geometry
  • The intersection of two subvarieties is well-defined when they intersect transversely
• In symplectic topology, the concept of transversality is generalized to the notion of symplectic transversality, which is important in the study of pseudoholomorphic curves
• Sard's theorem and transversality have analogs in complex analytic geometry, where they are used to study the generic behavior of holomorphic maps between complex manifolds