Characteristic classes
Introduction
Characteristic classes are a fundamental concept in the field of algebraic topology, a branch of mathematics that studies topological spaces with algebraic methods. These classes provide a way to associate algebraic invariants to vector bundles, which are collections of vector spaces parameterized by a topological space. Characteristic classes play a crucial role in understanding the geometry and topology of manifolds, as well as in various applications across mathematics and theoretical physics.
Historical Background
The development of characteristic classes can be traced back to the work of Hermann Weyl and Élie Cartan in the early 20th century. They laid the groundwork for the study of differential geometry and Lie groups, which are essential for understanding characteristic classes. The formal introduction of characteristic classes was made by Hassler Whitney and Eduard Stiefel in the 1930s and 1940s. Whitney introduced what are now known as Stiefel-Whitney classes, while Stiefel developed the concept of Stiefel manifolds, which are closely related to these classes.
Definition and Basic Properties
Characteristic classes are defined for vector bundles over a topological space. A vector bundle is a collection of vector spaces, one for each point in the space, that varies continuously from point to point. The most common types of characteristic classes include Stiefel-Whitney classes, Chern classes, Pontryagin classes, and Euler classes.
Stiefel-Whitney Classes
Stiefel-Whitney classes are defined for real vector bundles and take values in mod 2 cohomology. They are denoted by \( w_i \), where \( i \) is a non-negative integer. The total Stiefel-Whitney class is the sum of all \( w_i \). These classes are used to study the orientability and embedding properties of manifolds.
Chern Classes
Chern classes are defined for complex vector bundles and take values in integral cohomology. They are denoted by \( c_i \), where \( i \) is a non-negative integer. The total Chern class is the sum of all \( c_i \). Chern classes are crucial in the study of complex manifolds and holomorphic vector bundles.
Pontryagin Classes
Pontryagin classes are defined for real vector bundles and take values in integral cohomology. They are denoted by \( p_i \), where \( i \) is a non-negative integer. Pontryagin classes are used to study the topology of smooth manifolds and are related to the Gauss-Bonnet theorem.
Euler Classes
Euler classes are associated with oriented vector bundles and take values in integral cohomology. The Euler class is a top-dimensional characteristic class and is used to study the topology of manifolds, particularly in relation to the Poincaré-Hopf theorem.
Applications
Characteristic classes have numerous applications in mathematics and physics. In topology, they are used to distinguish between different types of vector bundles and to study the properties of manifolds. In differential geometry, characteristic classes are used to study curvature and other geometric properties. In theoretical physics, they play a role in the study of gauge theory and string theory.
Topological Applications
In topology, characteristic classes are used to classify vector bundles up to isomorphism. They provide invariants that can distinguish between bundles that are not equivalent. For example, the Thom isomorphism theorem uses characteristic classes to relate the cohomology of a manifold to the cohomology of its tangent bundle.
Geometric Applications
In differential geometry, characteristic classes are used to study the curvature of manifolds. The Chern-Weil theory provides a method for computing characteristic classes using curvature forms. This theory is essential for understanding the relationship between geometry and topology.
Physical Applications
In theoretical physics, characteristic classes are used in the study of gauge theories, which describe the fundamental forces of nature. They are also used in string theory, where they play a role in the study of D-branes and other topological objects.
Computation of Characteristic Classes
The computation of characteristic classes involves various techniques from algebraic topology and differential geometry. One common method is the use of spectral sequences, which provide a way to compute cohomology groups and characteristic classes. Another method is the use of Chern-Weil theory, which relates characteristic classes to curvature forms.
Spectral Sequences
Spectral sequences are a powerful tool in algebraic topology that provide a systematic way to compute cohomology groups. They are used to compute characteristic classes by relating the cohomology of a vector bundle to the cohomology of its base space.
Chern-Weil Theory
Chern-Weil theory provides a method for computing characteristic classes using curvature forms. This theory is based on the observation that certain combinations of curvature forms are closed and represent cohomology classes. These classes are the characteristic classes of the bundle.
Advanced Topics
Characteristic classes are a rich area of study with many advanced topics. These include the study of equivariant cohomology, K-theory, and cobordism theory.
Equivariant Cohomology
Equivariant cohomology is a generalization of ordinary cohomology that takes into account group actions on topological spaces. It is used to study characteristic classes in the presence of symmetries.
K-Theory
K-theory is a branch of algebraic topology that studies vector bundles using Grothendieck groups. It provides a framework for understanding characteristic classes in terms of vector bundle isomorphism classes.
Cobordism Theory
Cobordism theory studies manifolds up to cobordism, a relation that identifies manifolds that can be transformed into each other by adding or removing boundaries. Characteristic classes play a role in cobordism theory by providing invariants that distinguish between different cobordism classes.