# Chern character

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.