Difference between revisions of "Chern character"
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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 | ||
− | + | $$ | |
+ | \mathop{\rm ch} : K ^ {1} ( X) = {\widetilde{K} } {} ^ {0} ( S X ^ {+} ) | ||
+ | \rightarrow H ^ {\textrm{ odd } } ( X ; \mathbf Q ) | ||
+ | $$ | ||
coincide with the composition | coincide with the composition | ||
− | + | $$ | |
+ | \mathop{\rm ch} : {\widetilde{K} } {} ^ {0} ( S X ^ {+} ) \rightarrow \widetilde{H} {} ^ | ||
+ | { \mathop{\rm ev} } ( S X | ||
+ | ^ {+} ; \mathbf Q ) \rightarrow ^ { S- } 1 \widetilde{H} {} ^ {\textrm{ odd } } | ||
+ | ( X ^ {+} ; \mathbf Q ) = H ^ {\textrm{ odd } } ( X ; \mathbf Q ) | ||
+ | $$ | ||
− | (here "+" denotes the functor from the category of topological spaces into the category of pointed spaces | + | (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 | + | 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 | is defined by the formula | ||
− | + | $$ | |
+ | \sigma h ( \xi ) = e ^ {g ( \sigma _ {i} ( \xi ) ) } , | ||
+ | $$ | ||
− | where | + | 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 | ||
− | + | $$ | |
+ | \sigma h : K ^ {*} \rightarrow h ^ {**} ( X) \otimes \mathbf Q . | ||
+ | $$ | ||
− | For a generalized cohomology theory | + | 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 | 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 | + | 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 | + | 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|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.
Chern character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chern_character&oldid=14027