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
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.
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