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The '''Chern classes''' are a sequence of [[characteristic class]]es associated to complex [[vector bundle]]s, 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.

[[Category:Mathematics]] [[Category:Topology]] [[Category:Geometry]]