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A characteristic class defining a ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220201.png" />. For a one-dimensional bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220202.png" /> there is the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220204.png" /> is the rational [[Chern class|Chern class]]. This identity, together with the requirement that the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220205.png" /> define a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220206.png" />, uniquely determines the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c0220207.png" />. There is a commutative diagram
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<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/c022020/c0220208.png" /></td> </tr></table>
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{{TEX|done}}
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 +
A characteristic class defining a ring homomorphism  $  \mathop{\rm ch} : K ( X) \rightarrow H  ^ {**} ( X ; \mathbf Q ) $.
 +
For a one-dimensional bundle  $  \xi $
 +
there is the identity  $  \mathop{\rm ch}  \xi = e ^ {c _ {1} ( \xi ) } $,
 +
where  $  c _ {1} ( \xi ) $
 +
is the rational [[Chern class|Chern class]]. This identity, together with the requirement that the class  $  \mathop{\rm ch} $
 +
define a homomorphism  $  K  ^ {0} ( X) \rightarrow H ^ { \mathop{\rm ev} } ( X ;  \mathbf Q ) $,
 +
uniquely determines the class  $  \mathop{\rm ch} $.
 +
There is a commutative diagram
 +
 
 +
$$
 +
 
 +
\begin{array}{ccc}
 +
\mathop{\rm ch} : {\widetilde{K}  } {}  ^ {0} ( X)  &\rightarrow  & \widetilde{H}  {}  ^ {**} ( X ; \mathbf Q )  \\
 +
\downarrow  &{}  &\downarrow  \\
 +
\mathop{\rm ch} : {\widetilde{K}  } {}  ^ {0} ( S  ^ {2} \wedge X )  &\rightarrow  &\widetilde{H}  {}  ^ {**} ( S  ^ {2} \wedge X ; \mathbf Q ) ,  \\
 +
\end{array}
 +
 
 +
$$
  
 
in which the vertical arrows denote the periodicity operator and the dual [[Suspension|suspension]]. Let the mapping
 
in which the vertical arrows denote the periodicity operator and the dual [[Suspension|suspension]]. Let the mapping
  
<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/c022020/c0220209.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ch} : K  ^ {1} ( X)  = {\widetilde{K}  } {}  ^ {0} ( S X  ^ {+} )
 +
\rightarrow  H ^ {\textrm{ odd } }  ( X ; \mathbf Q )
 +
$$
  
 
coincide with the composition
 
coincide with the composition
  
<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/c022020/c02202010.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ch} : {\widetilde{K}  } {}  ^ {0} ( S X  ^ {+} )  \rightarrow  \widetilde{H}  {} ^
 +
{ \mathop{\rm ev} } ( S X
 +
^ {+} ; \mathbf Q )  \rightarrow ^ { S- }  1  \widetilde{H}  {} ^ {\textrm{ odd } }
 +
( X  ^ {+} ; \mathbf Q )  = H ^ {\textrm{ odd } }  ( X ;  \mathbf Q )
 +
$$
  
(here  "+"  denotes the functor from the category of topological spaces into the category of pointed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202011.png" />. One obtains a functorial transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202012.png" />, and this induces a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202013.png" />, which is a natural isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202014.png" />-graded rings.
+
(here  "+"  denotes the functor from the category of topological spaces into the category of pointed spaces $  X  ^ {+} = ( X \cup x _ {0} , x _ {0} ) $.  
 +
One obtains a functorial transformation $  \mathop{\rm ch} : K  ^ {*} ( X) \rightarrow H  ^ {**} ( X ;  \mathbf Q ) $,  
 +
and this induces a transformation $  K  ^ {*} ( X) \otimes \mathbf Q \rightarrow H  ^ {**} ( X ;  \mathbf Q ) $,  
 +
which is a natural isomorphism of $  \mathbf Z _ {2} $-
 +
graded rings.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202015.png" /> is a generalized cohomology theory in which the Chern classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202016.png" /> are defined, then for one-dimensional bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202017.png" /> the generalized Chern character
+
If $  h  ^ {*} $
 +
is a generalized cohomology theory in which the Chern classes $  \sigma _ {i} $
 +
are defined, then for one-dimensional bundles $  \xi $
 +
the generalized Chern character
  
<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/c022020/c02202018.png" /></td> </tr></table>
+
$$
 +
\sigma h ( \xi )  \in  h  ^ {**} ( X) \otimes \mathbf Q
 +
$$
  
 
is defined by the formula
 
is defined by the formula
  
<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/c022020/c02202019.png" /></td> </tr></table>
+
$$
 +
\sigma h ( \xi )  = e ^ {g ( \sigma _ {i} ( \xi ) ) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202020.png" /> is the logarithm of the [[Formal group|formal group]] corresponding to the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202021.png" />. By the splitting lemma one can define a natural ring homomorphism
+
where $  g ( t) $
 +
is the logarithm of the [[Formal group|formal group]] corresponding to the theory $  h  ^ {*} $.  
 +
By the splitting lemma one can define a natural ring homomorphism
  
<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/c022020/c02202022.png" /></td> </tr></table>
+
$$
 +
\sigma h : K  ^ {*}  \rightarrow  h  ^ {**} ( X) \otimes \mathbf Q .
 +
$$
  
For a generalized cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202023.png" /> there exists a unique natural isomorphism of graded groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202024.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202025.png" /> coincides with the mapping
+
For a generalized cohomology theory $  h  ^ {*} $
 +
there exists a unique natural isomorphism of graded groups $  \mathop{\rm ch} _ {h} : h  ^ {*} ( X) \rightarrow {\mathcal H}  ^ {**} ( X ;  h  ^ {*} (  \mathop{\rm pt} ) \otimes \mathbf Q ) $,  
 +
which for $  X = \mathop{\rm pt} $
 +
coincides with the mapping
  
<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/c022020/c02202026.png" /></td> </tr></table>
+
$$
 +
h  ^ {*} (  \mathop{\rm pt} )  \rightarrow  h  ^ {*} (  \mathop{\rm pt} ) \otimes \mathbf Q ,\ \
 +
x  \rightarrow  x \otimes 1 .
 +
$$
  
 
Here
 
Here
  
<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/c022020/c02202027.png" /></td> </tr></table>
+
$$
 +
[ {\mathcal H}  ^ {*} ( X ; h  ^ {*} (  \mathop{\rm pt} ) \otimes \mathbf Q ) ] _ {n}  = \
 +
\sum _ { i }  {\mathcal H}  ^ { i } ( X ; h  ^ {n-} i (  \mathop{\rm pt} ) \otimes \mathbf Q ) .
 +
$$
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202029.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202030.png" />-graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202031.png" />-theory, coincides with the Chern character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202032.png" />. The natural transformation functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202033.png" /> is called the Chern–Dold character.
+
The mapping $  \mathop{\rm ch} _ {k} $,  
 +
where $  K  ^ {*} $
 +
is a $  \mathbf Z _ {2} $-
 +
graded $  K $-
 +
theory, coincides with the Chern character $  \mathop{\rm ch} $.  
 +
The natural transformation functor $  \mathop{\rm ch} _ {h  ^ {*}  } $
 +
is called the Chern–Dold character.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202034.png" /> be the unitary cobordism theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202035.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202036.png" /> be the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202037.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202038.png" /> is isomorphic to the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202041.png" /> is the orientation of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202042.png" />. Analogously, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202043.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202045.png" /> is the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202046.png" />. The formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022020/c02202047.png" /> is the functional inverse of the Mishchenko series
+
Let $  h  ^ {*} $
 +
be the unitary cobordism theory $  U  ^ {*} $
 +
and let $  X $
 +
be the space $  \mathbf C P  ^  \infty  $.  
 +
The ring $  U  ^ {**} ( \mathbf C P  ^  \infty  ) $
 +
is isomorphic to the ring of formal power series $  \Omega _ {u}  ^ {*} [ [ u ] ] $,  
 +
where $  \Omega _ {u}  ^ {*} = U (  \mathop{\rm pt} ) $
 +
and $  u \in U  ^ {2} ( \mathbf C P  ^  \infty  ) $
 +
is the orientation of the bundle $  \kappa _ {1} $.  
 +
Analogously, the ring $  {\mathcal H}  ^ {*} ( \mathbf C P  ^  \infty  ;  \Omega _ {u}  ^ {*} ) $
 +
is isomorphic to $  \Omega _ {u}  ^ {*} [ [ x ] ] $,  
 +
where $  x \in H  ^ {2} ( \mathbf C P  ^  \infty  ) $
 +
is the orientation of $  \kappa _ {1} $.  
 +
The formal power series $  \mathop{\rm ch} _ {u} ( u) $
 +
is the functional inverse of the Mishchenko series
  
<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/c022020/c02202048.png" /></td> </tr></table>
+
$$
 +
g ( u)  = \sum _ { n= } 0 ^  \infty 
 +
 
 +
\frac{[ \mathbf C P  ^ {n} ] }{n+}
 +
1 u  ^ {n+} 1 .
 +
$$
  
 
For references see [[Chern class|Chern class]].
 
For references see [[Chern class|Chern class]].
 
 
  
 
====Comments====
 
====Comments====
 
Cf. the comments to [[Chern class|Chern class]] and [[Chern number|Chern number]].
 
Cf. the comments to [[Chern class|Chern class]] and [[Chern number|Chern number]].

Revision as of 16:43, 4 June 2020


A characteristic class defining a ring homomorphism $ \mathop{\rm ch} : K ( X) \rightarrow H ^ {**} ( X ; \mathbf Q ) $. For a one-dimensional bundle $ \xi $ there is the identity $ \mathop{\rm ch} \xi = e ^ {c _ {1} ( \xi ) } $, where $ c _ {1} ( \xi ) $ is the rational Chern class. This identity, together with the requirement that the class $ \mathop{\rm ch} $ define a homomorphism $ K ^ {0} ( X) \rightarrow H ^ { \mathop{\rm ev} } ( X ; \mathbf Q ) $, uniquely determines the class $ \mathop{\rm ch} $. There is a commutative diagram

$$ \begin{array}{ccc} \mathop{\rm ch} : {\widetilde{K} } {} ^ {0} ( X) &\rightarrow & \widetilde{H} {} ^ {**} ( X ; \mathbf Q ) \\ \downarrow &{} &\downarrow \\ \mathop{\rm ch} : {\widetilde{K} } {} ^ {0} ( S ^ {2} \wedge X ) &\rightarrow &\widetilde{H} {} ^ {**} ( S ^ {2} \wedge X ; \mathbf Q ) , \\ \end{array} $$

in which the vertical arrows denote the periodicity operator and the dual suspension. Let the mapping

$$ \mathop{\rm ch} : K ^ {1} ( X) = {\widetilde{K} } {} ^ {0} ( S X ^ {+} ) \rightarrow H ^ {\textrm{ odd } } ( X ; \mathbf Q ) $$

coincide with the composition

$$ \mathop{\rm ch} : {\widetilde{K} } {} ^ {0} ( S X ^ {+} ) \rightarrow \widetilde{H} {} ^ { \mathop{\rm ev} } ( S X ^ {+} ; \mathbf Q ) \rightarrow ^ { S- } 1 \widetilde{H} {} ^ {\textrm{ odd } } ( X ^ {+} ; \mathbf Q ) = H ^ {\textrm{ odd } } ( X ; \mathbf Q ) $$

(here "+" denotes the functor from the category of topological spaces into the category of pointed spaces $ X ^ {+} = ( X \cup x _ {0} , x _ {0} ) $. One obtains a functorial transformation $ \mathop{\rm ch} : K ^ {*} ( X) \rightarrow H ^ {**} ( X ; \mathbf Q ) $, and this induces a transformation $ K ^ {*} ( X) \otimes \mathbf Q \rightarrow H ^ {**} ( X ; \mathbf Q ) $, which is a natural isomorphism of $ \mathbf Z _ {2} $- graded rings.

If $ h ^ {*} $ is a generalized cohomology theory in which the Chern classes $ \sigma _ {i} $ are defined, then for one-dimensional bundles $ \xi $ the generalized Chern character

$$ \sigma h ( \xi ) \in h ^ {**} ( X) \otimes \mathbf Q $$

is defined by the formula

$$ \sigma h ( \xi ) = e ^ {g ( \sigma _ {i} ( \xi ) ) } , $$

where $ g ( t) $ is the logarithm of the formal group corresponding to the theory $ h ^ {*} $. By the splitting lemma one can define a natural ring homomorphism

$$ \sigma h : K ^ {*} \rightarrow h ^ {**} ( X) \otimes \mathbf Q . $$

For a generalized cohomology theory $ h ^ {*} $ there exists a unique natural isomorphism of graded groups $ \mathop{\rm ch} _ {h} : h ^ {*} ( X) \rightarrow {\mathcal H} ^ {**} ( X ; h ^ {*} ( \mathop{\rm pt} ) \otimes \mathbf Q ) $, which for $ X = \mathop{\rm pt} $ coincides with the mapping

$$ h ^ {*} ( \mathop{\rm pt} ) \rightarrow h ^ {*} ( \mathop{\rm pt} ) \otimes \mathbf Q ,\ \ x \rightarrow x \otimes 1 . $$

Here

$$ [ {\mathcal H} ^ {*} ( X ; h ^ {*} ( \mathop{\rm pt} ) \otimes \mathbf Q ) ] _ {n} = \ \sum _ { i } {\mathcal H} ^ { i } ( X ; h ^ {n-} i ( \mathop{\rm pt} ) \otimes \mathbf Q ) . $$

The mapping $ \mathop{\rm ch} _ {k} $, where $ K ^ {*} $ is a $ \mathbf Z _ {2} $- graded $ K $- theory, coincides with the Chern character $ \mathop{\rm ch} $. The natural transformation functor $ \mathop{\rm ch} _ {h ^ {*} } $ is called the Chern–Dold character.

Let $ h ^ {*} $ be the unitary cobordism theory $ U ^ {*} $ and let $ X $ be the space $ \mathbf C P ^ \infty $. The ring $ U ^ {**} ( \mathbf C P ^ \infty ) $ is isomorphic to the ring of formal power series $ \Omega _ {u} ^ {*} [ [ u ] ] $, where $ \Omega _ {u} ^ {*} = U ( \mathop{\rm pt} ) $ and $ u \in U ^ {2} ( \mathbf C P ^ \infty ) $ is the orientation of the bundle $ \kappa _ {1} $. Analogously, the ring $ {\mathcal H} ^ {*} ( \mathbf C P ^ \infty ; \Omega _ {u} ^ {*} ) $ is isomorphic to $ \Omega _ {u} ^ {*} [ [ x ] ] $, where $ x \in H ^ {2} ( \mathbf C P ^ \infty ) $ is the orientation of $ \kappa _ {1} $. The formal power series $ \mathop{\rm ch} _ {u} ( u) $ is the functional inverse of the Mishchenko series

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

For references see Chern class.

Comments

Cf. the comments to Chern class and Chern number.

How to Cite This Entry:
Chern character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_character&oldid=14027
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article