Chern character

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.