Characteristic class

Characteristic classes are defined for principal bundles and vector bundles over a topological space. For a vector bundle E → B, a characteristic class is a natural transformation from the functor of vector bundles to the cohomology ring of the base, satisfying a naturality condition under pullback. The most common construction uses the classifying space BO(n) or BU(n), where universal bundles live; characteristic classes are then pullbacks of cohomology classes of these spaces.¹ For example, the Stiefel–Whitney classes w_i(E) lie in Hⁱ(B; Z/2), while Chern classes c_i(E) lie in H²ⁱ(B; Z). These classes satisfy axioms such as naturality, the Whitney sum formula, and normalization for the tautological line bundle.

The four classical families of characteristic classes are the Stiefel–Whitney classes, Chern classes, Pontryagin classes, and the Euler class. Stiefel–Whitney classes are defined for real vector bundles and take values in mod 2 cohomology; they detect orientability and spin structures. Chern classes apply to complex vector bundles and are integral cohomology classes; they are used in the classification of complex line bundles via the first Chern class, which is an isomorphism to H²(B; Z). Pontryagin classes are defined for real bundles using complexification, and the Euler class is the top-dimensional obstruction to a nowhere-zero section. For example, the tangent bundle of the 2-sphere has Euler class equal to twice the generator of H²(S²), reflecting the Poincaré–Hopf theorem.²

Characteristic classes are indispensable in geometry and topology. In the theory of manifolds, the Stiefel–Whitney numbers and Pontryagin numbers are used to determine whether a manifold is cobordant to zero, leading to the classification of smooth manifolds up to cobordism. The Hirzebruch signature theorem expresses the signature of a manifold in terms of Pontryagin classes, and the Atiyah–Singer index theorem generalizes this to elliptic operators, where the topological index is given by characteristic classes.³ In algebraic geometry, Chern classes are used to define the Chern character and Todd class, which appear in the Grothendieck–Riemann–Roch theorem. In physics, characteristic classes appear in gauge theory, where instantons are classified by the second Chern number, and in the quantum Hall effect, where the first Chern number determines the Hall conductance.

Beyond the classical families, there are many other characteristic classes, such as the Wu classes, which are defined in terms of Steenrod squares and are related to Stiefel–Whitney classes via the Wu formula. The Chern–Simons forms, which are secondary characteristic classes, arise when primary classes vanish and are used in the study of 3-manifold invariants and in theoretical physics. The theory of characteristic classes extends to generalized cohomology theories, such as K-theory, where the Chern character provides an isomorphism between K-theory and rational cohomology. A subtle point is that characteristic classes are not complete invariants: for example, two non-isomorphic vector bundles can have the same Stiefel–Whitney and Chern classes. The classification of bundles is refined by Postnikov towers and obstruction theory, which use cohomology operations beyond characteristic classes.⁴

Fiber bundle

A space that locally looks like a product of a base space and a fiber, but may be globally twisted.

Cohomology

A sequence of abelian groups associated to a topological space, capturing global algebraic invariants.

Stiefel–Whitney class

A characteristic class for real vector bundles, taking values in mod 2 cohomology.

Chern class

A characteristic class for complex vector bundles, taking values in integral cohomology.

Euler class

A characteristic class that is the obstruction to a nowhere-zero section of an oriented vector bundle.