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A [[Characteristic class|characteristic class]] defined for complex vector bundles. A Chern class of the complex vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220301.png" /> over a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220302.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220303.png" /> and is defined for all natural indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220304.png" />. By the complete Chern class is meant the inhomogeneous characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220305.png" />, and the Chern polynomial is the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220307.png" /> is a formal unknown. Chern classes were introduced in [[#References|[1]]].
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The characteristic classes, defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220308.png" />-dimensional complex vector bundles and with values in the integral cohomology, are naturally identified with the elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c0220309.png" />. In this sense the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203010.png" /> can be thought of as elements of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203011.png" />, the complete Chern class as an element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203012.png" />, and the Chern polynomial as an element of the formal power series ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203013.png" />.
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The Chern classes satisfy the following properties, which uniquely determine them. 1) For two vector bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203014.png" /> with a common base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203016.png" />, in other words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203017.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203018.png" />. 2) For the one-dimensional universal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203020.png" /> the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203021.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203022.png" /> is the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203024.png" /> is the [[Thom space|Thom space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203025.png" />, which, being complex, has a uniquely-defined orientation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203026.png" />).
+
A [[Characteristic class|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 [[#References|[1]]].
  
Consequences of the properties 1)–2) are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203030.png" /> is the trivial bundle. The latter fact allows one to define Chern classes as elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203031.png" />.
+
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 ] ] $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203032.png" /> is a collection of non-negative integers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203033.png" /> denotes the characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203035.png" />.
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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|Thom space]] of  $  \kappa _ {1} $,
 +
which, being complex, has a uniquely-defined orientation  $  u $).
  
Under the natural monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203036.png" /> induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203037.png" />, the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203038.png" />. The image of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203040.png" /> is the subring consisting of all symmetric formal power series. Every symmetric formal power series in the Wu generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203041.png" /> determines a characteristic class that can be expressed in terms of Chern classes. For example, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203042.png" /> determines a characteristic class with rational coefficients, called the Todd class and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203043.png" />.
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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} ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203044.png" /> be a set of non-negative integers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203045.png" /> denote the characteristic class defined by the smallest symmetric polynomial in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203047.png" />, containing the monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203048.png" />.
+
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} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203049.png" /> be an oriented multiplicative cohomology theory. Then the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203050.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203051.png" /> satisfy, as do ordinary Chern classes, the properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203055.png" /> is the orientation of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203056.png" />, and these properties completely determine them. As with ordinary Chern classes, one uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203059.png" /> are two complex vector bundles, then
+
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 ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203060.png" /></td> </tr></table>
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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} } $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203061.png" /></td> </tr></table>
+
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
  
where the summation is taken over all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203063.png" />.
+
$$
 +
S _  \omega  ( \sigma _ {1} \dots \sigma _ {n} ) ( \xi \oplus \eta ) =
 +
$$
  
In place of the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203064.png" /> one may take a unitary [[Cobordism|cobordism]] theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203065.png" /> or [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203066.png" />-theory]]. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203067.png" />-theory the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203068.png" /> is defined by the identity mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203069.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203070.png" />-theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203072.png" /> is the Bott periodicity operator. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203073.png" /> is retained for Chern classes with values in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203074.png" />-theory, while Chern classes with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203075.png" />-theory are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203076.png" />.
+
$$
 +
= \
 +
\sum _ {\omega  ^  \prime  \cup \omega  ^ {\prime\prime} = \omega } S _ {
 +
\omega  ^  \prime  } ( \sigma _ {1} \dots \sigma _ {n} ) ( \xi ) S _ {\omega  ^ {\prime\prime}  } ( \sigma _ {1} \dots \sigma _ {n} ) ( \eta ) ,
 +
$$
  
According to the general theory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203078.png" /> is a vector bundle with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203079.png" />. However <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203080.png" />-theory is often conveniently thought of as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203081.png" />-graded theory, identifying the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203083.png" /> via the periodicity operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203084.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203086.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203087.png" />. From this point of view it makes sense to consider, instead of the complete Chern class, the Chern polynomial
+
where the summation is taken over all sets  $  \omega  ^  \prime  , \omega  ^ {\prime\prime} $
 +
with $  \omega  ^  \prime  \cup \omega  ^ {\prime\prime} = \omega $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203088.png" /></td> </tr></table>
+
In place of the theory  $  h  ^ {*} $
 +
one may take a unitary [[Cobordism|cobordism]] theory  $  U  ^ {*} $
 +
or [[K-theory| $  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} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203089.png" /> be a cohomology operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203090.png" />-theory (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203091.png" /> terms). The polynomial
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203092.png" /></td> </tr></table>
+
$$
 +
\gamma _ {t} ( \xi )  = 1 + \sum _ {i > 0 }
 +
\gamma _ {i} ( \xi ) t  ^ {i}  \in  K  ^ {0} ( B) [ t] .
 +
$$
  
satisfies, as does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203093.png" />, the multiplicative property
+
Let  $  \lambda  ^ {i} ( \xi ) = [ \xi \wedge \dots \wedge \xi ] $
 +
be a cohomology operation in  $  K $-
 +
theory ( $  i $
 +
terms). The polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203094.png" /></td> </tr></table>
+
$$
 +
\lambda _ {t} ( \xi )  = \sum _ { i= } 0 ^  \infty 
 +
\lambda  ^ {i} ( \xi ) t  ^ {i}  \in  K  ^ {0} ( B) [ t]
 +
$$
 +
 
 +
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:
 
There is the following connection between these polynomials:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203095.png" /></td> </tr></table>
+
$$
 +
 
 +
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203097.png" /> is the trivial bundle of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203098.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c02203099.png" /> in this construction are different from those constructed by M.F. Atiyah, who defined them by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030100.png" />. R. Stong [[#References|[2]]] defined classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030101.png" /> that satisfy the condition
+
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 [[#References|[2]]] defined classes $  \gamma _ {i} $
 +
that satisfy the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030102.png" /></td> </tr></table>
+
$$
 +
\gamma _ {t} ( \xi )  =
 +
\frac{\lambda _ {t} }{1-}
 +
t
 +
( \overline \xi \; - \mathop{\rm dim}  \xi ) .
 +
$$
  
 
The difference arises because, for Stong,
 
The difference arises because, for Stong,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030103.png" /></td> </tr></table>
+
$$
 +
= \beta ( [ \kappa _ {1} ] - 1 )  \in \
 +
\widetilde{K}  {}  ^ {2} ( \mathbf C P  ^  \infty  ) .
 +
$$
 +
 
 +
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|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|cohomology operation]] in the  $  U  ^ {*} $-
 +
theory. The subalgebra of the [[Steenrod algebra|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  $.
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030104.png" /> are connected with the notion of a Landweber–Novikov algebra, which is very fruitful in homotopy theory. For an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030105.png" /> of non-negative integers, consider the characteristic class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030107.png" />. There is a [[Thom isomorphism|Thom isomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030109.png" /> is the spectrum corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030110.png" />-theory. The image of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030111.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030112.png" /> determines a [[Cohomology operation|cohomology operation]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030113.png" />-theory. The subalgebra of the [[Steenrod algebra|Steenrod algebra]] in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030114.png" />-theory generated by the operations of this form is called the Landweber–Novikov algebra. The operation constructed from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030115.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030116.png" />.
+
For one-dimensional bundles  $  \xi , \eta $
 +
there is the identity
  
For one-dimensional bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030117.png" /> there is the identity
+
$$
 +
c _ {1} ( \xi \otimes \eta )  = c _ {1} ( \xi ) + c _ {1} ( \eta ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030118.png" /></td> </tr></table>
+
This important property, which enables one to define the [[Chern character|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
  
This important property, which enables one to define the [[Chern character|Chern character]], does not hold in generalized cohomology theories. However there exists a formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030119.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030120.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030121.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030122.png" /> is the first Chern class with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030123.png" />. For the unitary cobordism theory
+
$$
 +
g ( t)  = \sum _ { n= } 0 ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030124.png" /></td> </tr></table>
+
\frac{[ \mathbf C P  ^ {n} ] }{n+}
 +
1 t  ^ {n+} 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030125.png" /> is the cobordism class of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030126.png" />. This series is called the Mishchenko series.
+
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====
 
====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 {{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>
 
<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>
 
 
  
 
====Comments====
 
====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030128.png" /> denotes the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030129.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030130.png" />.
+
$  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030131.png" /> for a complex oriented cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030132.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030133.png" /> is the logarithm of the formal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030134.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022030/c022030135.png" />; cf. [[Cobordism|Cobordism]] and [[Formal group|Formal group]] for some more details.
+
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|Cobordism]] and [[Formal group|Formal group]] for some more details.

Revision as of 16:43, 4 June 2020


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.

How to Cite This Entry:
Chern class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_class&oldid=46333
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article