Chern character

A characteristic class defining a ring homomorphism . For a one-dimensional bundle there is the identity , where is the rational Chern class. This identity, together with the requirement that the class define a homomorphism , uniquely determines the class . There is a commutative diagram

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

coincide with the composition

(here "+" denotes the functor from the category of topological spaces into the category of pointed spaces . One obtains a functorial transformation , and this induces a transformation , which is a natural isomorphism of -graded rings.

If is a generalized cohomology theory in which the Chern classes are defined, then for one-dimensional bundles the generalized Chern character

is defined by the formula

where is the logarithm of the formal group corresponding to the theory . By the splitting lemma one can define a natural ring homomorphism

For a generalized cohomology theory there exists a unique natural isomorphism of graded groups , which for coincides with the mapping

Here

The mapping , where is a -graded -theory, coincides with the Chern character . The natural transformation functor is called the Chern–Dold character.

Let be the unitary cobordism theory and let be the space . The ring is isomorphic to the ring of formal power series , where and is the orientation of the bundle . Analogously, the ring is isomorphic to , where is the orientation of . The formal power series is the functional inverse of the Mishchenko series

For references see Chern class.

Comments

Cf. the comments to Chern class and Chern number.