Chern class

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 [1].

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 [2] 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.

References

[1]	S.S. Chern, "Characteristic classes of Hermitian manifolds" Ann. of Math. , 47 : 1 (1946) pp. 85–121
[2]	R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[3]	R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965)
[4]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964)
[5a]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" Ann. of Math. (2) , 87 (1968) pp. 484–530
[5b]	M.F. Atiyah, G.B. Segal, "The index of elliptic operators II" Ann. of Math. (2) , 87 (1968) pp. 531–545
[5c]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" Ann. of Math. (2) , 87 (1968) pp. 546–604
[5d]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators IV" Ann. of Math. (2) , 93 (1971) pp. 119–138
[5e]	M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" Ann. of Math. (2) , 93 (1971) pp. 139–149
[6]	F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German)
[7]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[8]	V.M. Bukhshtaber, "The Chern–Dold character in cobordisms" Math. USSR-Sb. , 12 : 4 (1970) pp. 573–594 Mat. Sb. , 83 (1970) pp. 575–595
[9]	S.P. Novikov, "The method of algebraic topology from the viewpoint of cobordism theory" Math. USSR-Izv. , 4 : 1 (1967) pp. 827–913 Izv. Akad. SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951
[10]	M.F. Atiyah, "-theory: lectures" , Benjamin (1967)

Comments

denotes the completion of .

The power series for a complex oriented cohomology theory such that is the logarithm of the formal group defined by ; cf. Cobordism and Formal group for some more details.