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Chern character

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Idea

The Chern character of a generalized (Eilenberg-Steenrod) cohomology theory is a canonical morphism from the generalized cohomology to ordinary cohomology. When thought of in the refinement to differential cohomology and thinking of a cocycle in differential cohomology as a generalization of a connection on a bundle, the Chern-character is the map that sends a generalized connection to its curvature characteristic form.

The archetypical example is the Chern-character from complex K-theory to ordinary cohomology. (This is a map from a complex oriented cohomology theory of chromatic level 1 to chromatic level 0. More generally one can also consider higher chromatic Chern characters that take values not in ordinary cohomology but in some cohomology theory of higher chromatic level. See at higher chromatic Chern character for more on this.)

More in detail, for every generalized (Eilenberg-Steenrod) cohomology theory given by a spectrum EE, there is a canonical natural isomorphism from the rationalized EE-cohomology to ordinary (Eilenberg-MacLane) cohomology with coefficients in the rationalized homotopy groups of EE:

H(,E)H(,(π *E)). H(-,E)\otimes \mathbb{R} \;\; \stackrel{\simeq}{\to} \;\; H(-,(\pi_* E)\otimes \mathbb{R}) \,.

So rationally, every generalized (Eilenberg-Steenrod) cohomology is ordinary (Eilenberg-MacLane) cohomology. Hence every generalized (ES-)cohomology theory has a canonical morphism to ordinary real cohomology

ch:H(,E)H(,E)H(,π *E). ch : H(-,E) \;\; \to \;\; H(-,E)\otimes \mathbb{R} \;\; \stackrel{\simeq}{\to} \;\; H(-,\pi_* E \otimes \mathbb{R}) \,.

In the case that E=KUE = KU is the K-theory spectrum, this morphism is classically known as the Chern character. Generalizing from this example, the term “Chern character” is sometimes used also for the general case.

There are analogues in algebraic geometry (e.g. a Chern character between the Chow groups and the algebraic K-theory) and noncommutative geometry (Chern-Connes character) where the role of usual cohomology is taken by some variant of cyclic cohomology.

For vector bundles and K-theory

The classical theory of the Chern character applies to the spectrum of complex K-theory, E=KUE = KU. In this case, the Chern character is made up from Chern classes: each characteristic class is by Chern-Weil theory in image of certain element in the Weil algebra via taking the class of evaluation at the curvature operator for some choice of a connection. Consider the symmetric functions in nn variables t 1,,t nt_1,\ldots, t_n and let the Chern classes of a complex vector bundle ξ\xi (representing a complex K-theory class) are c 1,,c nc_1,\ldots, c_n. Define the formal power series

ϕ=ϕ n(t 1,,t n)=e t 1++e t n= k=0 1k!(t 1 k++t n k) \phi = \phi^n(t_1,\ldots, t_n) = e^{t_1}+\ldots+e^{t_n}= \sum_{k=0}^\infty \frac{1}{k!} (t_1^k+\ldots+t_n^k)

Then ch(χ)=ϕ(c 1,,c n)ch(\chi) = \phi(c_1,\ldots,c_n).

Let us describe this a bit differently. The cocycle H 0(X,KU)H^0(X,KU) may be represented by a complex vector bundle, and the image of this cocycle under the Chern-character is the class in even-graded real cohomology that is represented (under the deRham theorem isomorphism of deRham cohomology with real cohomology) by the even graded closed differential form

ch():= jk jtr(F F )Ω 2(X), ch(\nabla) := \sum_{j \in \mathbb{N}} k_j tr( F_\nabla \wedge \cdots \wedge F_\nabla) \;\; \in \Omega^{2 \bullet}(X) \,,

where

  • \nabla is any chosen connection on the vector bundle;

  • F=F Ω 2(X,End(V))F = F_\nabla \in \Omega^2(X,End(V)) is the curvature of this connection;

  • k jk_j \in \mathbb{R} are normalization constants, k j=1j!(12πi) jk_j = \frac{1}{j!} \left( \frac{1}{2\pi i}\right)^j;

  • the trace of the wedge products produces the curvature characteristic forms.

The Chern character applied to the Whitney sum of two vector bundles is a sum of the Chern characters for the two: ch(ξη)=ch(χ)+ch(η)ch(\xi\oplus \eta) = ch(\chi)+ch(\eta) and it is multiplicative under the tensor product of vector bundles: ch(ξη)=ch(χ)ch(η)ch(\xi\otimes\eta)=ch(\chi)ch(\eta). Therefore we get a ring homomorphism.

General theory

The fundamental cocycle

The isomorphism

H(,E)H(,(π *E)) H(-,E)\otimes \mathbb{R} \;\; \stackrel{\simeq}{\to} \;\; H(-,(\pi_* E)\otimes \mathbb{R})

that defines the Chern-character map is induced by a canonical cocycle on the spectrum EE that is called the fundamental cocycle.

This is described for instance in section 4.8, page 47 of Hopkins-Singer Quadratic Functions in Geometry, Topology,and M-Theory.

Properties

Push-forward

The behaviour of the Chern-character under fiber integration in generalized cohomology along proper maps is described by the Grothendieck-Riemann-Roch theorem.

References

The universal Chern character for generalized (Eilenberg-Steenrod) cohomology theory is discussed in section 4.8, page 47 of

A characterization of Chern-character maps for K-theory is in

A discussion of Chern characters in terms of free loop space objects in derived geometry is in

which conjectures a construction that is fully developed in

See also

  • Marc Hoyois, Chern character and derived algebraic geometry (2009) (pdf)

Revised on November 7, 2013 02:56:10 by Urs Schreiber (77.251.114.72)