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

$$g ( t) = \sum _ { n= } 0 ^ \infty \frac{[ \mathbf C P ^ {n} ] }{n+} 1 t ^ {n+} 1 ,$$

where $[ \mathbf C P ^ {n} ] = \Omega _ {u} ^ {*} = U ^ {*} ( \mathop{\rm pt} )$ is the cobordism class of the projective space $\mathbf C P ^ {n}$. 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, "-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.