Editing Chern class (section)

== Geometric approach ==

=== Basic idea and motivation ===

Chern classes are [[characteristic class]]es. They are [[topological invariant]]s associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.

In topology, differential geometry, and algebraic geometry, it is often important to count how many [[linearly independent]] sections a vector bundle has. The Chern classes offer some information about this through, for instance, the [[Riemann–Roch theorem]] and the [[Atiyah–Singer index theorem]].

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the [[curvature form]].

=== Construction ===

There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.

The original approach to Chern classes was via algebraic topology: the Chern classes arise via [[homotopy theory]] which provides a mapping associated with a vector bundle to a [[classifying space]] (an infinite [[Grassmannian]] in this case). For any complex vector bundle ''V'' over a manifold ''M'', there exists a map ''f'' from ''M'' to the classifying space such that the bundle ''V'' is equal to the pullback, by ''f'', of a universal bundle over the classifying space, and the Chern classes of ''V'' can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of [[Schubert cycle]]s.

It can be shown that for any two maps ''f'', ''g'' from ''M'' to the classifying space whose pullbacks are the same bundle ''V'', the maps must be homotopic. Therefore, the pullback by either ''f'' or ''g'' of any universal Chern class to a cohomology class of ''M'' must be the same class.  This shows that the Chern classes of ''V'' are well-defined.

Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the [[Chern–Weil theory]].

There is also an approach of [[Alexander Grothendieck]] showing that axiomatically one need only define the line bundle case.

Chern classes arise naturally in [[algebraic geometry]].  The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, [[locally free sheaves]]) over any nonsingular variety.  Algebro-geometric Chern classes do not require the underlying field to have any special properties.  In particular, the vector bundles need not necessarily be complex.

Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a [[Section (category theory)|section]] of a vector bundle: for example the theorem saying one can't comb a hairy ball flat ([[hairy ball theorem]]).  Although that is strictly speaking a question about a ''real'' vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.

See [[Chern–Simons theory]] for more discussion.