🧬Cohomology Theory Unit 9 – Characteristic classes

Key Concepts and Definitions

• Characteristic classes assign cohomology classes to vector bundles over a topological space
• Provide a way to measure the "twisting" or non-triviality of a vector bundle
• Cohomology classes are elements of the cohomology ring of the base space
  • Cohomology ring is a graded ring that encodes topological information
• Examples of characteristic classes include Chern classes, Pontryagin classes, and Stiefel-Whitney classes
• Characteristic classes are natural transformations from the functor of vector bundles to the cohomology functor
• Satisfy certain axioms such as naturality, additivity, and normalization
• Used to study the topology of manifolds and their embeddings into Euclidean spaces

Historical Context and Development

• Characteristic classes were introduced by Eduard Stiefel and Hassler Whitney in the 1930s
• Developed as a tool to study the topology of vector bundles and manifolds
• Chern classes, named after Shiing-Shen Chern, were introduced in the 1940s for complex vector bundles
• Pontryagin classes, named after Lev Pontryagin, were introduced in the 1940s for real vector bundles
• The Hirzebruch-Riemann-Roch theorem (1950s) relates characteristic classes to the index of elliptic operators
• The Atiyah-Singer index theorem (1960s) generalizes the Hirzebruch-Riemann-Roch theorem
  • Connects characteristic classes, K-theory, and the index of elliptic operators
• Characteristic classes have played a crucial role in the development of algebraic topology and differential geometry

Types of Characteristic Classes

• Chern classes are characteristic classes for complex vector bundles
  • Denoted by $c_i(E)$, where $E$ is a complex vector bundle and $i$ is the degree
  • The total Chern class is defined as $c(E) = 1 + c_1(E) + c_2(E) + \cdots$
• Pontryagin classes are characteristic classes for real vector bundles
  • Denoted by $p_i(E)$, where $E$ is a real vector bundle and $i$ is the degree
  • The total Pontryagin class is defined as $p(E) = 1 + p_1(E) + p_2(E) + \cdots$
• Stiefel-Whitney classes are characteristic classes for real vector bundles
  • Denoted by $w_i(E)$, where $E$ is a real vector bundle and $i$ is the degree
  • The total Stiefel-Whitney class is defined as $w(E) = 1 + w_1(E) + w_2(E) + \cdots$
• Euler class is a characteristic class for oriented real vector bundles
  • Measures the obstruction to the existence of a nowhere-vanishing section
• Other characteristic classes include the Thom class, the Wu class, and the Chern character

Computation Techniques

• Characteristic classes can be computed using various techniques depending on the context
• The splitting principle reduces the computation of characteristic classes to the case of line bundles
  • Allows for the use of the Whitney product formula and the Cartan formula
• The Chern-Weil theory computes characteristic classes using differential forms and curvature
  • Expresses characteristic classes as polynomials in the curvature form of a connection
• The Grothendieck-Riemann-Roch theorem computes characteristic classes in algebraic geometry
  • Relates the Chern character of a coherent sheaf to its pushforward under a proper morphism
• The Atiyah-Hirzebruch spectral sequence computes characteristic classes using cellular cohomology
• The Serre spectral sequence computes characteristic classes of fiber bundles
• Characteristic classes can also be computed using classifying spaces and homotopy theory

Applications in Topology

• Characteristic classes are powerful tools for studying the topology of manifolds and vector bundles
• The non-vanishing of characteristic classes can detect the non-triviality of vector bundles
  • For example, a non-zero Euler class implies that the vector bundle has no nowhere-vanishing section
• Characteristic classes can obstruct the existence of certain structures on manifolds
  • The Stiefel-Whitney classes obstruct the existence of spin structures and orientation
  • The Pontryagin classes obstruct the existence of almost complex structures
• The Chern classes are related to the complex structure and the Dolbeault cohomology of complex manifolds
• Characteristic classes can be used to define and study characteristic numbers of manifolds
  • For example, the Euler characteristic and the signature can be expressed using characteristic classes
• The Hirzebruch signature theorem relates the signature of a manifold to its Pontryagin classes
• Characteristic classes play a crucial role in the classification of manifolds and vector bundles

Connections to Other Mathematical Areas

• Characteristic classes have deep connections to various branches of mathematics
• In algebraic topology, characteristic classes are related to K-theory and cobordism theory
  • The Chern character provides a ring homomorphism from K-theory to rational cohomology
• In differential geometry, characteristic classes are related to curvature and the geometry of connections
  • The Gauss-Bonnet theorem relates the Euler characteristic to the curvature of a Riemannian manifold
• In algebraic geometry, characteristic classes are related to the Chow ring and intersection theory
  • The Grothendieck-Riemann-Roch theorem computes characteristic classes of coherent sheaves
• In physics, characteristic classes appear in gauge theory and string theory
  • The Chern classes are related to the quantization of magnetic charge and the classification of instantons
• Characteristic classes have applications in combinatorics and graph theory
  • The Stiefel-Whitney classes of a matroid can be used to study its combinatorial properties
• The theory of characteristic classes has inspired the development of generalized cohomology theories

Advanced Topics and Current Research

• Equivariant characteristic classes study vector bundles with group actions
  • Extend the theory of characteristic classes to the equivariant setting
• Quantum characteristic classes are a generalization of characteristic classes to the quantum cohomology ring
  • Arise in the study of quantum cohomology and Gromov-Witten theory
• Characteristic classes of foliations and Lie algebroids extend the theory to more general geometric structures
• The theory of characteristic classes has been generalized to other cohomology theories
  • For example, the Morava K-theory and the elliptic cohomology
• The relationship between characteristic classes and homotopy theory is an active area of research
  • Homotopy invariance and the classification of vector bundles using classifying spaces
• Characteristic classes have applications in mathematical physics and string theory
  • For example, the Witten genus and the elliptic genus are related to characteristic classes
• The study of characteristic classes in the context of derived algebraic geometry and higher category theory

Problem-Solving Strategies

• Identify the type of vector bundle (real, complex, oriented) and the base space
• Determine the appropriate characteristic classes to use based on the context
  • For example, Chern classes for complex bundles, Stiefel-Whitney classes for real bundles
• Use the properties of characteristic classes (naturality, additivity, Whitney product formula) to simplify computations
• Apply the splitting principle to reduce computations to the case of line bundles
• Utilize the Chern-Weil theory to compute characteristic classes using differential forms and curvature
  • Express the characteristic classes as polynomials in the curvature form of a connection
• Use the Grothendieck-Riemann-Roch theorem to compute characteristic classes in algebraic geometry
• Apply spectral sequences (Atiyah-Hirzebruch, Serre) to compute characteristic classes in specific situations
• Relate characteristic classes to other topological invariants (Euler characteristic, signature) using classical theorems
• Interpret the vanishing or non-vanishing of characteristic classes in terms of the geometry of the vector bundle or manifold
• Use characteristic classes to obstruct the existence of certain structures (spin structures, almost complex structures)
• Apply characteristic classes to classify vector bundles and manifolds up to isomorphism or diffeomorphism