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Difference between revisions of "Chern character"

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m (tex encoded by computer)
m (fixing superscripts)
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\begin{array}{ccc}
 
\begin{array}{ccc}
  \mathop{\rm ch} :  {\widetilde{K}  } {}  ^ {0} ( X)  &\rightarrow  & \widetilde{H}  {}  ^ {**} ( X ;  \mathbf Q )  \\
+
  \mathop{\rm ch} :  {\widetilde{K}  }  ^ {0} ( X)  &\rightarrow  & \widetilde{H}  ^ {**} ( X ;  \mathbf Q )  \\
 
\downarrow  &{}  &\downarrow  \\
 
\downarrow  &{}  &\downarrow  \\
  \mathop{\rm ch} :  {\widetilde{K}  } {}  ^ {0} ( S  ^ {2} \wedge X )  &\rightarrow  &\widetilde{H} {}  ^ {**} ( S  ^ {2} \wedge X ;  \mathbf Q ) ,  \\
+
  \mathop{\rm ch} :  {\widetilde{K}  }  ^ {0} ( S  ^ {2} \wedge X )  &\rightarrow  &\widetilde{H} ^ {**} ( S  ^ {2} \wedge X ;  \mathbf Q ) ,  \\
 
\end{array}
 
\end{array}
  
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$$  
 
$$  
  \mathop{\rm ch} :  K  ^ {1} ( X)  =  {\widetilde{K}  } {}  ^ {0} ( S X  ^ {+} )
+
  \mathop{\rm ch} :  K  ^ {1} ( X)  =  {\widetilde{K}  }  ^ {0} ( S X  ^ {+} )
 
  \rightarrow  H ^ {\textrm{ odd } }  ( X ;  \mathbf Q )
 
  \rightarrow  H ^ {\textrm{ odd } }  ( X ;  \mathbf Q )
 
$$
 
$$
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$$  
 
$$  
  \mathop{\rm ch} :  {\widetilde{K}  } {}  ^ {0} ( S X  ^ {+} )  \rightarrow  \widetilde{H}  {} ^  
+
  \mathop{\rm ch} :  {\widetilde{K}  }  ^ {0} ( S X  ^ {+} )  \rightarrow  \widetilde{H}  {} ^  
 
{ \mathop{\rm ev} } ( S X
 
{ \mathop{\rm ev} } ( S X
  ^ {+} ;  \mathbf Q )  \rightarrow ^ { S- }  1  \widetilde{H}  {} ^ {\textrm{ odd } }
+
  ^ {+} ;  \mathbf Q )  \rightarrow ^ { S- }  1  \widetilde{H}  ^ {\textrm{ odd } }
 
( X  ^ {+} ;  \mathbf Q )  =  H ^ {\textrm{ odd } }  ( X ;  \mathbf Q )
 
( X  ^ {+} ;  \mathbf Q )  =  H ^ {\textrm{ odd } }  ( X ;  \mathbf Q )
 
$$
 
$$
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One obtains a functorial transformation  $  \mathop{\rm ch} :  K  ^ {*} ( X) \rightarrow H  ^ {**} ( X ;  \mathbf Q ) $,  
 
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 ) $,  
 
and this induces a transformation  $  K  ^ {*} ( X) \otimes \mathbf Q \rightarrow H  ^ {**} ( X ;  \mathbf Q ) $,  
which is a natural isomorphism of  $  \mathbf Z _ {2} $-
+
which is a natural isomorphism of  $  \mathbf Z _ {2} $-graded rings.
graded rings.
 
  
 
If  $  h  ^ {*} $
 
If  $  h  ^ {*} $
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$$  
 
$$  
 
[ {\mathcal H}  ^ {*} ( X ;  h  ^ {*} (  \mathop{\rm pt} ) \otimes \mathbf Q ) ] _ {n}  = \  
 
[ {\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 ) .
+
\sum _ { i }  {\mathcal H}  ^ { i } ( X ;  h  ^ {n-i} (  \mathop{\rm pt} ) \otimes \mathbf Q ) .
 
$$
 
$$
  
 
The mapping  $  \mathop{\rm ch} _ {k} $,  
 
The mapping  $  \mathop{\rm ch} _ {k} $,  
 
where  $  K  ^ {*} $
 
where  $  K  ^ {*} $
is a  $  \mathbf Z _ {2} $-
+
is a  $  \mathbf Z _ {2} $-graded  $  K $-theory, coincides with the Chern character  $  \mathop{\rm ch} $.  
graded  $  K $-
 
theory, coincides with the Chern character  $  \mathop{\rm ch} $.  
 
 
The natural transformation functor  $  \mathop{\rm ch} _ {h  ^ {*}  } $
 
The natural transformation functor  $  \mathop{\rm ch} _ {h  ^ {*}  } $
 
is called the Chern–Dold character.
 
is called the Chern–Dold character.
Line 117: Line 114:
  
 
$$  
 
$$  
g ( u)  =  \sum _ { n= } 0 ^  \infty   
+
g ( u)  =  \sum _ { n= 0} ^  \infty   
  
\frac{[ \mathbf C P  ^ {n} ] }{n+}
+
\frac{[ \mathbf C P  ^ {n} ] }{n+1} u  ^ {n+1} .
1 u  ^ {n+} 1 .
 
 
$$
 
$$
  

Revision as of 11:50, 21 March 2022


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=46332
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article