nLab
Chern character

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential cohomology

Contents

Idea

Traditionally, in the strict sense of the term, the Chern character is a universal characteristic class of vector bundles or equivalently of their topological K-theory classes, which is a rational combination of all Chern classes.

This is a special case of the following more general construction (Hopkins-Singer 02, section 4.8):

for EE a spectrum representing a generalized (Eilenberg-Steenrod) cohomology theory there is a canonical localization map

ch E:EEH ch_E \;\colon\; E \longrightarrow E \wedge H\mathbb{R}

to the smash product with the Eilenberg-MacLane spectrum over the real numbers. This represents the EE-Chern character (see also Bunke-Gepner 13, around def. 2.1).

In the case that E=E = KU this reproduces the traditional Chern character. (In which case 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.)

The Chern character ch Ech_E may be used to define differential cohomology refinements E^\hat E of the cohomology theory EE by choosing a differential form-model for EHE \wedge H\mathbb{R} (Hopkins-Singer 02, see also at differential function complex). In that case ch Ech_E is the real cohomology class associated to a chern character differential form CH ECH_E via the de Rham theorem. Here CH ECH_E has the interpretation of being the curvature forms of the differential cohomology cocycles thought of as ∞-connections.

This may be turned around (Bunke-Nikolaus-Völkl 13, prop. .3.5): given any refinement E^\hat E of EE in a tangent cohesive (∞,1)-topos THT \mathbf{H}, then it is induced from homotopy pullback of its de Rham coefficients along a Chern character map

ch E=Πθ E^:EΠ(E^)Π dRE^, ch_E = \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E \,,

where Π\Pi is the shape modality and θ E\theta_E the Maurer-Cartan form of EE. This reproduces the above definition for ordinary differential form models, see at differential cohomology diagram – Hopkins-Singer coefficients.

But more generally, given for instance a K(n)-localization EL K(n)EE \longrightarrow L_{K(n)} E then any choice of cohesive refinement of L K(n)EL_{K(n)} E (i.e. lift through the unit of the shape modality Π\Pi) which is in the kernel of \flat yields a generalized differential cohomology theory E^\hat E whose intrinsic Chern-character Πθ E^\Pi \theta_{\hat E} is the K(n)K(n)-localization. See at differential cohomology diagram – Chern character and differential fracture.

In words this is summarized succintly as: The Chern character is the shape of the Maurer-Cartan form.

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

Examples

For vector bundles and topological 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 the image of a 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) be 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()coloeqq jk jtr(F F )Ω 2(X), ch(\nabla) \coloeqq \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.

For spectra and generalized cohomology theories

For EE a spectrum and E E^\bullet the generalized cohomology theory it represents

E (X)π Maps(X,E) E^\bullet(X) \;\simeq\; \pi_{-\bullet} Maps(X,E)

the Chern-Dold character for EE (Buchstaber 70) is the map induced by rationalization over the real numbers

EL E E \overset{L_{\mathbb{R}}}{\longrightarrow} E_{\mathbb{R}}

i.e. is

(1)chd:E (X)π Maps(X,E)π Maps(X,L )π Maps(X,E )E 𝔼 (X)H (X,π (E) ). chd \;\colon\; E^\bullet(X) \;\simeq\; \pi_{-\bullet}Maps(X,E) \overset{ \pi_{-\bullet}Maps(X,L_{\mathbb{R}}) }{\longrightarrow} \pi_{-\bullet}Maps(X,E_{\mathbb{R}}) \;\simeq\; E^\bullet_{\mathbb{E}}(X) \;\simeq\; H^\bullet(X, \pi_{\bullet}(E)\otimes_{\mathbb{Z}}\mathbb{R}) \,.

The very last equivalence in (1) is due to Dold 56, Cor. 4 (reviewed in detail in Rudyak 98, II.3.17, see also Gross 19, Def. 2.5).

One place where this neat state of affairs (1) is made fully explicit is Lind-Sati-Westerland 16, Def. 2.1. Many other references leave this statement somewhat in between the lines (e.g. Buchstaber 70, Upmeier 14) and, in addition, often without reference to Dold (e.g. Hopkins-Singer 02, Sec. 4.8, Bunke 12, Def. 4.45, Bunke-Gepner 13, Def. 2.1, Bunke-Nikolaus 14, p. 17).

Beware that some authors say Chern-Dold character for the full map in (1) (e.g. Buchstaber 70, Upmeier 14, Lind-Sati-Westerland 16, Def. 2.1), while other authors mean by this only that last equivalence in (1) (e.g. Rudyak 98, II.3.17, Gross 19, Def. 2.5).

Examples of Chern-Dold characters:

For cohesive stable homotopy types

More generally, for E^\hat E a stable homotopy type in a cohesive (∞,1)-topos, then the underlying bare homotopy type is EΠ(E^)E \coloneqq \Pi(\hat E) and the corresponding Chern character is

chΠθ E^:EΠ(E^)Π dRE^. ch \coloneqq \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E \,.

For more on this see at differential cohomology diagram.

In terms of cyclic homology

Generalizing in another direction, generalized Chern characters are given by passage to derived loop spaces and their cyclic homology or, more generally, topological cyclic homology (Toen-Vezzosi 08, Hoyois-Scherotzke-Sibilla 15).

Properties

Push-forward and Grothendieck-Riemann-Roch theorem

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

Compatibility with the Adams operations

The Adams operations ψ k\psi^k on complex topological K-theory are compatible with the Chern character map to rational cohomology in that the effect of ψ k\psi^k on the Chern character image in degree 2r2r is multiplication by k rk^r:

Definition

(Adams-like operations on rational cohomology)

For XX a topological space, with rational cohomology in even degrees denoted

H ev(X;):rH 2r(X;) H^{ev}(X;\, \mathbb{Q}) \;\colon\; \underset{r \in \mathbb{N}}{\prod} H^{2 r}(X;\, \mathbb{Q})

define graded linear maps

ψ H k:H ev(X)H ev(X) \psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)

for kk \in \mathbb{N} by taking their restriction to degree 2r2r to act by multiplication with k rk^r:

H 2r(X;) ψ H k H 2r(X;) α 2k k rα. \array{ H^{2r}(X;\mathbb{Q}) &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}& H^{2r}(X;\mathbb{Q}) \\ \alpha_{2k} &\mapsto& k^{r} \cdot \alpha \,. }
Proposition

(Adams operations compatible with the Chern character)

For XX a topological space with a finite CW-complex-mathematical structure, the Chern character chch on the complex topological K-theory of XX intertwines the Adams operations ψ n\psi^n on K-theory with the Adams-like operations ψ H n\psi^n_H on rational cohomology from Def. , for k1k \geq 1, in that the following diagram commutes:

K(X) ch H ev(X;) ψ k ψ H k K(X) ch H ev(X;), \array{ K(X) &\overset{\;\;\;ch\;\;\;}{\longrightarrow}& H^{ev}(X;\,\mathbb{Q}) \\ {}^{ \mathllap{ \psi^k } } \big\downarrow && \big\downarrow {}^{ \mathrlap{ \psi^k_H } } \\ K(X) &\underset{\;\;\;ch\;\;\;}{\longrightarrow}& H^{ev}(X;\,\mathbb{Q}) \,, }
Proof

Use the exponentional-formula for the Chern character with the splitting principle.

Detailed review is Maakestad 06, Thm. 4.9

References

Chern character on K-theory

Original Discussion of the Chern character on complex topological K-theory:

Further discussion

Discussion of the equivariant Chern character in equivariant K-theory:

  • German Stefanich, Chern Character in Twisted and Equivariant K-Theory (pdf)

Chern-Dold character on generalized cohomology

The identification of rational generalized cohomology as ordinary cohomology with coefficients in the rationalized stable homotopy groups is due to

  • Albrecht Dold, Relations between ordinary and extraordinary homology, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 (mathnet:mat350), reprinted in: J. Adams & G. Shepherd (Authors), Algebraic Topology: A Student’s Guide (London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press 1972 (doi:10.1017/CBO9780511662584.015)

reviewed in

The combination of Dold 56 to the Chern-Dold character on generalized (Eilenberg-Steenrod) cohomology theory is due (for complex cobordism cohomology) to

Review in

That the Chern-Dold character reduces to the original Chern character on K-theory is

That the Chern-Dold character is given by rationalization of representing spectra is made fully explicit in

This rationalization construction appears also (without attribution to #Hilton 71 or Buchstaber 70 or Dold 56) in the following articles (all in the context of differential cohomology):

More on the Chern-Dold character on complex cobordism cohomology:

The observation putting this into the general context of differential cohomology diagrams (see there) of stable homotopy types in cohesion is due to

based on Bunke-Gepner 13.

Further generalization of the Chern-Dold character to non-abelian cohomology:

The equivariant Chern-Dold character in equivariant cohomology:

In derived algebraic geometry

Discussion of Chern characters in terms of free loop space objects in derived geometry:

which conjectures a construction that is fully developed in

See also

Last revised on November 29, 2020 at 15:48:30. See the history of this page for a list of all contributions to it.