Chern class

A characteristic class defined for complex vector bundles. A Chern class of the complex vector bundle $ \xi $ over a base $ B $ is denoted by $ c _ {i} ( \xi ) \in H ^ {2i} ( B) $ and is defined for all natural indices $ i $. By the complete Chern class is meant the inhomogeneous characteristic class $ 1 + c _ {1} + c _ {2} + \dots $, and the Chern polynomial is the expression $ c _ {t} = 1 + c _ {1} t + c _ {2} t ^ {2} + \dots $, where $ t $ is a formal unknown. Chern classes were introduced in [1].

The characteristic classes, defined for all $ n $- dimensional complex vector bundles and with values in the integral cohomology, are naturally identified with the elements of the ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) $. In this sense the Chern classes $ c _ {i} $ can be thought of as elements of the groups $ H ^ {2i} ( \mathop{\rm BU} _ {n} ) $, the complete Chern class as an element of the ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) $, and the Chern polynomial as an element of the formal power series ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) [ [ t ] ] $.

The Chern classes satisfy the following properties, which uniquely determine them. 1) For two vector bundles $ \xi , \eta $ with a common base $ B $, $ c ( \xi \oplus \eta ) = c ( \xi ) c ( \eta ) $, in other words $ c _ {k} ( \xi \oplus \eta ) = \sum _ {i} c _ {i} ( \xi ) c _ {k-i} ( \eta ) $ where $ c _ {0} = 1 $. 2) For the one-dimensional universal bundle $ \kappa _ {1} $ over $ \mathbf C P ^ \infty $ the identity $ c ( \kappa _ {1} ) = 1 + u $ holds, where $ u \in H ^ {2} ( \mathbf C P ^ \infty ) $ is the orientation of $ \kappa _ {1} $( $ \mathbf C P ^ \infty $ is the Thom space of $ \kappa _ {1} $, which, being complex, has a uniquely-defined orientation $ u $).

Consequences of the properties 1)–2) are: $ c _ {i} ( \xi ) = 0 $ for $ i > \mathop{\rm dim} \xi $, and $ c ( \xi ) = c ( \xi \oplus \theta ) $, where $ \theta $ is the trivial bundle. The latter fact allows one to define Chern classes as elements of the ring $ H ^ {**} ( \mathop{\rm BU} ) $.

If $ \omega = \{ i _ {1} \dots i _ {k} \} $ is a collection of non-negative integers, then $ c _ \omega $ denotes the characteristic class $ c _ {i _ {1} } \dots c _ {i _ {k} } \in H ^ {2n} ( \mathop{\rm BU} ) $, where $ n = i _ {1} + \dots + i _ {k} $.

Under the natural monomorphism $ H ^ {**} ( \mathop{\rm BU} _ {n} ) \rightarrow H ^ {**} ( \mathop{\rm BT} _ {n} ) = \mathbf Z [ [ x _ {1} \dots x _ {n} ] ] $ induced by the mapping $ \mathop{\rm BT} _ {n} = \mathbf C P ^ \infty \times \dots \times \mathbf C P ^ \infty \rightarrow \mathop{\rm BU} _ {n} $, the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial $ \prod _{i=1} ^ {n} ( 1 + x _ {i} ) $. The image of the ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) $ in $ H ^ {**} ( \mathop{\rm BT} _ {n} ) = \mathbf Z [ [ x _ {1} \dots x _ {n} ] ] $ is the subring consisting of all symmetric formal power series. Every symmetric formal power series in the Wu generators $ x _ {1} \dots x _ {n} $ determines a characteristic class that can be expressed in terms of Chern classes. For example, the series $ \prod _{i=1} ^ {n} x _ {i} / ( 1 - e ^ {x _ {i} } ) $ determines a characteristic class with rational coefficients, called the Todd class and denoted by $ T \in H ^ {**} ( \mathop{\rm BU} _ {n} ; \mathbf Q ) $.

Let $ \omega = \{ i _ {1} \dots i _ {k} \} $ be a set of non-negative integers. Let $ S _ \omega ( c _ {1} \dots c _ {n} ) $ denote the characteristic class defined by the smallest symmetric polynomial in the variables $ x _ {1} \dots x _ {n} $, where $ n \geq i _ {1} + \dots + i _ {k} $, containing the monomial $ x _ {1} ^ {i _ {1} } \dots x _ {k} ^ {i _ {k} } $.

Let $ h ^ {*} $ be an oriented multiplicative cohomology theory. Then the Chern classes $ \sigma _ {i} $ with values in $ h ^ {*} $ satisfy, as do ordinary Chern classes, the properties: $ \sigma ( \xi \oplus \eta ) = \sigma ( \xi ) \sigma ( \eta ) $, $ \sigma = 1 + \sigma _ {1} + \sigma _ {2} + \dots $, $ \sigma ( \kappa _ {1} ) = 1 + u \in h ^ {*} ( \mathbf C P ^ \infty ) $, where $ u \in h ^ {2} ( \mathbf C P ^ \infty ) $ is the orientation of the bundle $ \kappa _ {1} $, and these properties completely determine them. As with ordinary Chern classes, one uses the notation $ \sigma _ \omega = \sigma _ {i _ {1} } \dots \sigma _ {i _ {k} } $ and $ S _ \omega ( \sigma _ {1} \dots \sigma _ {n} ) $. If $ \xi , \eta $ are two complex vector bundles, then

$$ S _ \omega ( \sigma _ {1} \dots \sigma _ {n} ) ( \xi \oplus \eta ) = $$

$$ = \ \sum _ {\omega ^ \prime \cup \omega ^ {\prime\prime} = \omega } S _ { \omega ^ \prime } ( \sigma _ {1} \dots \sigma _ {n} ) ( \xi ) S _ {\omega ^ {\prime\prime} } ( \sigma _ {1} \dots \sigma _ {n} ) ( \eta ) , $$

where the summation is taken over all sets $ \omega ^ \prime , \omega ^ {\prime\prime} $ with $ \omega ^ \prime \cup \omega ^ {\prime\prime} = \omega $.

In place of the theory $ h ^ {*} $ one may take a unitary cobordism theory $ U ^ {*} $ or $ K $- theory. For a $ U ^ {*} $- theory the element $ u \in U ^ {2} ( \mathbf C P ^ \infty ) $ is defined by the identity mapping $ \mathbf C P ^ \infty \rightarrow \mathbf C P ^ \infty = \mathop{\rm MU} _ {1} $, and for $ K $- theory $ u = \beta ( 1 - [ \overline{x}\; ] ) \in \widetilde{K} {} ^ {2} ( \mathbf C P ^ \infty ) $, where $ \widetilde \beta : K ^ {0} \rightarrow K ^ {2} $ is the Bott periodicity operator. The notation $ \sigma _ {i} $ is retained for Chern classes with values in a $ U ^ {*} $- theory, while Chern classes with values in $ K $- theory are denoted by $ \gamma _ {i} $.

According to the general theory, $ \gamma _ {i} ( \xi ) \in K ^ {2i} ( B) $, where $ \xi $ is a vector bundle with base $ B $. However $ K $- theory is often conveniently thought of as a $ \mathbf Z _ {2} $- graded theory, identifying the groups $ K ^ {n} ( B) $ and $ K ^ {n+} 2 ( B) $ via the periodicity operator $ \beta $. Then $ K ^ {*} ( B) = K ^ {0} ( B) \oplus K ^ {1} ( B) $ and $ \gamma ( \xi ) \in K ^ {0} ( B) $ for all $ i $. From this point of view it makes sense to consider, instead of the complete Chern class, the Chern polynomial

$$ \gamma _ {t} ( \xi ) = 1 + \sum _ {i > 0 } \gamma _ {i} ( \xi ) t ^ {i} \in K ^ {0} ( B) [ t] . $$

Let $ \lambda ^ {i} ( \xi ) = [ \xi \wedge \dots \wedge \xi ] $ be a cohomology operation in $ K $- theory ( $ i $ terms). The polynomial

$$ \lambda _ {t} ( \xi ) = \sum _{i=0}^ \infty \lambda ^ {i} ( \xi ) t ^ {i} \in K ^ {0} ( B) [ t] $$

satisfies, as does $ \gamma _ {t} $, the multiplicative property

$$ \lambda _ {t} ( \xi \oplus \eta ) = \ \lambda _ {t} ( \xi ) \lambda _ {t} ( \eta ) . $$

There is the following connection between these polynomials:

$$ \frac{\lambda _ {t} }{1-t} ( \overline \xi \; - \mathop{\rm dim} \xi ) = 1 + \sum _{i=1} ^ \infty (- 1) \gamma _ {i} ( \xi ) t ^ {i} = \ \gamma _ {-t} ( \xi ) . $$

Here both parts of the equation lie in $ K ^ {0} ( B) [ t] $ and $ \xi $ is the trivial bundle of dimension $ \mathop{\rm dim} \xi $. The classes $ \gamma _ {i} $ in this construction are different from those constructed by M.F. Atiyah, who defined them by the formula $ \gamma _ {t} ( \xi ) = ( \lambda _ {t} / ( 1 - t ) ) ( \xi ) $. R. Stong [2] defined classes $ \gamma _ {i} $ that satisfy the condition

$$ \gamma _ {t} ( \xi ) = \frac{\lambda _ {t} }{1-t} ( \overline \xi \; - \mathop{\rm dim} \xi ) . $$

The difference arises because, for Stong,

$$ u = \beta ( [ \kappa _ {1} ] - 1 ) \in \ \widetilde{K} {} ^ {2} ( \mathbf C P ^ \infty ) . $$

The classes $ \sigma _ {i} $ are connected with the notion of a Landweber–Novikov algebra, which is very fruitful in homotopy theory. For an arbitrary set $ \omega = \{ i _ {1} \dots i _ {k} \} $ of non-negative integers, consider the characteristic class $ S _ \omega ( \sigma _ {1} \dots \sigma _ {n} ) \in U ^ {2d} ( \mathop{\rm BU} ) $, where $ d = i _ {1} + \dots + i _ {k} $. There is a Thom isomorphism $ U ^ {2d} ( \mathop{\rm BU} ) \rightarrow \widetilde{U} {} ^ {2d} ( \mathop{\rm MU} ) \subset U ^ {2d} ( \mathop{\rm MU} ) $, where $ \mathop{\rm MU} $ is the spectrum corresponding to the $ U ^ {*} $- theory. The image of the class $ S _ \omega ( \sigma _ {1} \dots \sigma _ {n} ) $ in $ U ^ {2d} ( \mathop{\rm MU} ) $ determines a cohomology operation in the $ U ^ {*} $- theory. The subalgebra of the Steenrod algebra in the $ U ^ {*} $- theory generated by the operations of this form is called the Landweber–Novikov algebra. The operation constructed from the set $ \omega = \{ i _ {1} \dots i _ {k} \} $ is denoted by $ S _ \omega $.

For one-dimensional bundles $ \xi , \eta $ there is the identity

$$ c _ {1} ( \xi \otimes \eta ) = c _ {1} ( \xi ) + c _ {1} ( \eta ) . $$

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 $ g ( t) $ with coefficients in $ h ^ {*} ( \mathop{\rm pt} ) \otimes \mathbf Q $, such that $ g ( \sigma _ {1} ( \xi \otimes \eta )) = g ( \sigma _ {1} ( \xi ) ) + g ( \sigma _ {1} ( \eta ) ) $, where $ \sigma _ {1} $ is the first Chern class with coefficients in $ h ^ {*} $. For the unitary cobordism theory

$$ g(t) = \sum_{n=0}^\infty \frac{[\CC P^{n}]}{n+1} t^{n+1}, $$

where $[\CC P^{n}] = \Omega_{u}^{*} = U^{*} (\mathop{\rm pt})$ is the cobordism class of the projective space $ \CC P ^ {n} $. 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, "K-theory: lectures" , Benjamin (1967) MR224083

Comments

$ H ^ {**} ( X) $ denotes the completion $ \prod _ {i \geq 0 } H ^ {i} ( X) $ of $ H ^ {*} ( X) = \oplus _ {i \geq 0 } H ^ {i} ( X) $.

The power series $ g ( t) \in h ^ {*} ( \mathop{\rm pt} ) \otimes \mathbf Q $ for a complex oriented cohomology theory $ h ^ {*} $ such that $ g ( \sigma _ {1} ( \xi \otimes \eta ) ) = g ( \sigma _ {1} ( \xi ) ) + g ( \sigma _ {1} ( \eta ) ) $ is the logarithm of the formal group $ F _ {h} ( X , Y ) $ defined by $ h ^ {*} $; cf. Cobordism and Formal group for some more details.