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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Chern,   "Characteristic classes of Hermitian manifolds" ''Ann. of Math.'' , '''47''' : 1 (1946) pp. 85–121</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.S. Palais,   "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer (1964)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators I" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 484–530</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> M.F. Atiyah,   G.B. Segal,   "The index of elliptic operators II" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 531–545</TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top"> M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators III" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 546–604</TD></TR><TR><TD valign="top">[5d]</TD> <TD valign="top"> M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators IV" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 119–138</TD></TR><TR><TD valign="top">[5e]</TD> <TD valign="top"> M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators V" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 139–149</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch,   "Topological methods in algebraic geometry" , Springer (1978) (Translated from German)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. Husemoller,   "Fibre bundles" , McGraw-Hill (1966)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.F. Atiyah,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030127.png" />-theory: lectures" , Benjamin (1967)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Chern, "Characteristic classes of Hermitian manifolds" ''Ann. of Math.'' , '''47''' : 1 (1946) pp. 85–121 {{MR|0015793}} {{ZBL|0060.41416}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) {{MR|0198494}} {{ZBL|0137.17002}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 484–530 {{MR|0236950}} {{MR|0232402}} {{ZBL|0164.24001}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> M.F. Atiyah, G.B. Segal, "The index of elliptic operators II" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 531–545 {{MR|0236953}} {{MR|0236951}} {{ZBL|0164.24201}} </TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" ''Ann. of Math. (2)'' , '''87''' (1968) pp. 546–604 {{MR|0236952}} {{ZBL|0164.24301}} </TD></TR><TR><TD valign="top">[5d]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators IV" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 119–138 {{MR|0279833}} {{ZBL|0212.28603}} </TD></TR><TR><TD valign="top">[5e]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 139–149 {{MR|0279834}} {{ZBL|0212.28603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) {{MR|1335917}} {{MR|0202713}} {{ZBL|0376.14001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|0219.57027}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030127.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR></table>
  
  

Revision as of 21:50, 30 March 2012

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 MR0015793 Zbl 0060.41416
[2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[3] R.S. Palais, "Seminar on the Atiyah–Singer index theorem" , Princeton Univ. Press (1965) MR0198494 Zbl 0137.17002
[4] P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[5a] M.F. Atiyah, I.M. Singer, "The index of elliptic operators I" Ann. of Math. (2) , 87 (1968) pp. 484–530 MR0236950 MR0232402 Zbl 0164.24001
[5b] M.F. Atiyah, G.B. Segal, "The index of elliptic operators II" Ann. of Math. (2) , 87 (1968) pp. 531–545 MR0236953 MR0236951 Zbl 0164.24201
[5c] M.F. Atiyah, I.M. Singer, "The index of elliptic operators III" Ann. of Math. (2) , 87 (1968) pp. 546–604 MR0236952 Zbl 0164.24301
[5d] M.F. Atiyah, I.M. Singer, "The index of elliptic operators IV" Ann. of Math. (2) , 93 (1971) pp. 119–138 MR0279833 Zbl 0212.28603
[5e] M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" Ann. of Math. (2) , 93 (1971) pp. 139–149 MR0279834 Zbl 0212.28603
[6] F. Hirzebruch, "Topological methods in algebraic geometry" , Springer (1978) (Translated from German) MR1335917 MR0202713 Zbl 0376.14001
[7] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[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 Zbl 0219.57027
[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) MR224083


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.

How to Cite This Entry:
Chern class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_class&oldid=15158
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article