Characteristic class

A characteristic class is a cohomology class assigned naturally to a vector bundle or related geometric structure. It captures invariants that can distinguish bundles and constrain their possible sections or geometric structures.

Vector bundle: A space assembled from vector spaces that vary continuously over a base space. Characteristic classes assign cohomological invariants to these families of vector spaces.
Chern class: An integral cohomology class associated with a complex vector bundle. Chern classes are the central family of characteristic classes for complex bundles.
Hairy ball theorem: The theorem that every continuous tangent vector field on an even-dimensional sphere has a zero. The Euler class explains the obstruction to a nowhere-vanishing tangent field.
Chern–Simons invariant: A secondary invariant associated with a connection whose change can detect characteristic-class data. Unlike characteristic classes, it depends on connection-level information and can vary continuously.
Cohomology: A collection of algebraic invariants that detects topological features of a space. Characteristic classes live in cohomology groups of the bundle’s base space.
Stiefel–Whitney class: A mod-two cohomology class associated with a real vector bundle. These classes detect real-bundle features, including orientability and obstructions to sections.
Gauss–Bonnet theorem: A theorem equating total curvature of a compact surface with its Euler characteristic. It identifies an integral of curvature with the Euler class of the tangent bundle.
K-theory: A generalized cohomology theory built from stable classes of vector bundles. K-theory classifies bundles in a richer structure, while characteristic classes map bundles into cohomology.
Classifying space: A space whose maps classify bundles of a specified type over suitable base spaces. Universal characteristic classes on classifying spaces generate classes for individual bundles.
Pontryagin class: An integral cohomology class associated with a real vector bundle, defined using its complexification. Pontryagin classes provide integral invariants for real bundles beyond mod-two data.

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