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Chern class

In algebraic topology, the Chern classes of a complex vector bundle V on a topological space X are defined in the theory of characteristic classes. They lie in the cohomology of X, in the even-dimensional spaces; if V is a line bundle there is just a single (first) Chern class in the second cohomology group of X.

There are various ways of approaching the subject: originally Chern used differential geometry, in algebraic topology the Chern classes arise via homotopy theory which provides a mapping associated to V to a classifying space (an infinitary Grassmannian in this case), and there is an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case. Chern classes also arise naturally in algebraic geometry.

The intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat.

The name is given for Shiing-shen Chern, who first gave a general definition. It was realised in retrospect that geometers had met these classes already in a number of guises.

See Chern-Simons for more discussion.

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