Chern class

The Chern classes are a sequence of characteristic classes associated to complex vector bundles, named after Shiing-Shen Chern. They measure how a bundle "twists" over the base space and provide the primary topological invariants for classifying complex bundles. The first Chern class c₁(E) of a line bundle E is particularly simple: it is the cohomology class of the curvature form of any connection on E, and it classifies line bundles up to smooth isomorphism.

For higher-rank bundles, the total Chern class c(E) = 1 + c₁(E) + c₂(E) + ... lives in the cohomology ring of the base manifold and satisfies a Whitney sum formula: c(E ⊕ F) = c(E) ∪ c(F). This multiplicativity makes Chern classes computable and powerful. In physics, c₁ encodes magnetic charge in gauge theory; in algebraic geometry, Chern classes appear in intersection theory and the Riemann-Roch theorem.

The Chern character — a formal power series in the Chern classes — provides a natural map from K-theory to cohomology and is the key ingredient in the Atiyah-Singer index theorem's topological index formula.