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
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in which the vertical arrows denote the periodicity operator and the dual suspension. Let the mapping
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coincide with the composition
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(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
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is defined by the formula
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where is the logarithm of the formal group corresponding to the theory
. By the splitting lemma one can define a natural ring homomorphism
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For a generalized cohomology theory there exists a unique natural isomorphism of graded groups
, which for
coincides with the mapping
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Here
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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
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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=46332