Chern class

Category:Homology theoryCategory:Algebraic topologyCategory:Differential geometryCategory:Differential topology In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are a particular type of characteristic class associated to complex vector bundles. Chern classes are named for Shiing-shen Chern, who first gave a general definition of them in the 1940s.

Table of contents

1 Properties of Chern classes 2 Chern classes of almost complex manifolds and cobordism 3 Definitions of Chern classes 4 Generalizations

Properties of Chern classes

Given a complex vector bundle V over a topological space X, \nthe Chern classes of V are a sequence of elements of the cohomology of X. The n-th Chern class of V, which is usually denoted c_n(V), is an element of

H²ⁿ(X;Z),

the cohomology of X with integer coefficients. The class c₀(V) is always equal to 1. When V is a bundle of complex dimension d, then the classes c_n are equal to 0 for n > d. For instance, if V is a line bundle there is just a single (first) Chern class in the second cohomology group of X. The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H²(X;Z), which associates to a line bundle its first Chern class. Addition in the second dimensional cohomology group coincides with tensor product of complex line bundles. For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.

Chern classes of almost complex manifolds and cobordism

The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds. If M is an almost complex manifold, then its tangent bundle is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental class of M, giving an integer, called a chern number of M. \nIf M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M.

Definitions of Chern classes

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 (hairy ball theorem), though that is strictly speaking a question about a real vector bundle. See Chern-Simons for more discussion.

Generalizations

There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory. The theories for which such generalization is possible are called complex orientable.\nThe formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a formal group law.