A characteristic class defined for complex vector bundles. A Chern class of the complex vector bundle over a base is denoted by and is defined for all natural indices . By the complete Chern class is meant the inhomogeneous characteristic class , and the Chern polynomial is the expression , where is a formal unknown. Chern classes were introduced in .
The characteristic classes, defined for all -dimensional complex vector bundles and with values in the integral cohomology, are naturally identified with the elements of the ring . In this sense the Chern classes can be thought of as elements of the groups , the complete Chern class as an element of the ring , and the Chern polynomial as an element of the formal power series ring .
The Chern classes satisfy the following properties, which uniquely determine them. 1) For two vector bundles with a common base , , in other words where . 2) For the one-dimensional universal bundle over the identity holds, where is the orientation of ( is the Thom space of , which, being complex, has a uniquely-defined orientation ).
Consequences of the properties 1)–2) are: for , and , where is the trivial bundle. The latter fact allows one to define Chern classes as elements of the ring .
If is a collection of non-negative integers, then denotes the characteristic class , where .
Under the natural monomorphism induced by the mapping , the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial . The image of the ring in is the subring consisting of all symmetric formal power series. Every symmetric formal power series in the Wu generators determines a characteristic class that can be expressed in terms of Chern classes. For example, the series determines a characteristic class with rational coefficients, called the Todd class and denoted by .
Let be a set of non-negative integers. Let denote the characteristic class defined by the smallest symmetric polynomial in the variables , where , containing the monomial .
Let be an oriented multiplicative cohomology theory. Then the Chern classes with values in satisfy, as do ordinary Chern classes, the properties: , , , where is the orientation of the bundle , and these properties completely determine them. As with ordinary Chern classes, one uses the notation and . If are two complex vector bundles, then
where the summation is taken over all sets with .
In place of the theory one may take a unitary cobordism theory or -theory. For a -theory the element is defined by the identity mapping , and for -theory , where is the Bott periodicity operator. The notation is retained for Chern classes with values in a -theory, while Chern classes with values in -theory are denoted by .
According to the general theory, , where is a vector bundle with base . However -theory is often conveniently thought of as a -graded theory, identifying the groups and via the periodicity operator . Then and for all . From this point of view it makes sense to consider, instead of the complete Chern class, the Chern polynomial
Let be a cohomology operation in -theory ( terms). The polynomial
satisfies, as does , the multiplicative property
There is the following connection between these polynomials:
Here both parts of the equation lie in and is the trivial bundle of dimension . The classes in this construction are different from those constructed by M.F. Atiyah, who defined them by the formula . R. Stong  defined classes that satisfy the condition
The difference arises because, for Stong,
The classes are connected with the notion of a Landweber–Novikov algebra, which is very fruitful in homotopy theory. For an arbitrary set of non-negative integers, consider the characteristic class , where . There is a Thom isomorphism , where is the spectrum corresponding to the -theory. The image of the class in determines a cohomology operation in the -theory. The subalgebra of the Steenrod algebra in the -theory generated by the operations of this form is called the Landweber–Novikov algebra. The operation constructed from the set is denoted by .
For one-dimensional bundles there is the identity
This important property, which enables one to define the Chern character, does not hold in generalized cohomology theories. However there exists a formal power series with coefficients in , such that , where is the first Chern class with coefficients in . For the unitary cobordism theory
where is the cobordism class of the projective space . This series is called the Mishchenko series.
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denotes the completion of .
Chern class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_class&oldid=15158