nLab Chern character

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

$$ch_E \;\colon\; E \longrightarrow E \wedge H\mathbb{R}$$

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

In the case that $E =$ 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 $ch_E$ may be used to define differential cohomology refinements $\hat E$ of the cohomology theory $E$ by choosing a differential form-model for $E \wedge H\mathbb{R}$ (Hopkins-Singer 02, see also at differential function complex). In that case $ch_E$ is the real cohomology class associated to a chern character differential form $CH_E$ via the de Rham theorem. Here $CH_E$ 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 $\hat E$ of $E$ in a tangent cohesive (∞,1)-topos $T \mathbf{H}$, then it is induced from homotopy pullback of its de Rham coefficients along a Chern character map

$$ch_E = \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E \,,$$

where $\Pi$ is the shape modality and $\theta_E$ the Maurer-Cartan form of $E$. 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 $E \longrightarrow L_{K(n)} E$ then any choice of cohesive refinement of $L_{K(n)} E$ (i.e. lift through the unit of the shape modality $\Pi$) which is in the kernel of $\flat$ yields a generalized differential cohomology theory $\hat E$ whose intrinsic Chern-character $\Pi \theta_{\hat E}$ is the $K(n)$-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, $E = KU$. 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 $n$ variables $t_1,\ldots, t_n$ and let the Chern classes of a complex vector bundle $\xi$ (representing a complex K-theory class) be $c_1,\ldots, c_n$. Define the formal power series

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

Then $ch(\chi) = \phi(c_1,\ldots,c_n)$.

Let us describe this a bit differently. The cocycle $H^0(X,KU)$ 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

$$ch(\nabla) \coloneqq \sum_{j \in \mathbb{N}} k_j tr( F_\nabla \wedge \cdots \wedge F_\nabla) \;\; \in \Omega^{2 \bullet}(X) \,,$$

where

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

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

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

• 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: $ch(\xi\oplus \eta) = ch(\chi)+ch(\eta)$ and it is multiplicative under the tensor product of vector bundles: $ch(\xi\otimes\eta)=ch(\chi)ch(\eta)$. Therefore we get a ring homomorphism.

For spectra and generalized cohomology theories

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

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

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

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

i.e. is

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

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:

• Chern character on KU;

• Pontrjagin character on KO.

For cohesive stable homotopy types

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

$$ch \coloneqq \Pi \theta_{\hat E} \;\colon\; E \simeq \Pi(\hat E) \longrightarrow \Pi \flat_{dR} \hat E \,.$$

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

Intertwining Thom classes with Todd classes

Proposition

(rational Todd class is Chern character of Thom class)

Let $V \to X$ be a complex vector bundle over a compact topological space. Then the Todd class $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in rational cohomology equals the Chern character $ch$ of the Thom class $th(V) \,\in\, K\big( Th(V) \big)$ in the complex topological K-theory of the Thom space $Th(V)$, when both are compared via the Thom isomorphisms $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$$\phi_{H\mathbb{Q}} \big( Td(V) \big) \;=\; ch\big( th(V) \big) \,.$$

More generally , for $x \in K(X)$ any class, we have

$$\phi_{H\mathbb{Q}} \big( ch(x) \cup Td(V) \big) \;=\; ch\big( \phi_{K}(x) \big) \,,$$

which specializes to the previous statement for $x = 1$.

(Karoubi 78, Chapter V, Theorem 4.4)

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

Definition

(Adams-like operations on rational cohomology)

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

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

define graded linear maps

$$\psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)$$

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

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

Proposition

(Adams operations compatible with the Chern character)

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

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

(Adams 62, Thm. 5.1. (vi), review in Karoubi 78, Chapter V, Theorem 3.27, Maakestad 06, Thm. 4.9)

Proof idea

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

Related concepts

• odd Chern character

• twisted Chern character

• equivariant Chern character

• twisted equivariant Chern character

• Chern-Dold character

• Bismut-Chern character

• for algebraic K-theory:

  • cyclotomic trace

  • regulator, Beilinson regulator

• Grothendieck-Riemann-Roch theorem

• higher chromatic Chern character

  • elliptic Chern character

  • Morava E-theoretic Chern character

• transchromatic character

• group character

References

Chern character on K-theory

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

• Friedrich Hirzebruch, Section 12.1 of: Neue topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)

• Armand Borel, Friedrich Hirzebruch, Section 9.1 in: Characteristic Classes and Homogeneous Spaces, I, American Journal of Mathematics Vol. 80, No. 2 (Apr., 1958), pp. 458-538 (jstor:2372795)

• M. F. Atiyah, F. Hirzebruch, Section 1.10 in: Vector bundles and homogeneous spaces, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. (web, doi:10.1142/9789814401319_0008, MR 0139181)

• Peter Hilton, General cohomology theory and K-theory, London Mathematical Society Lecture Note Series 1, Cambridge University Press (1971) (doi:10.1017/CBO9780511662577)

• Max Karoubi, Chapter V.3 of: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007%2F978-3-540-79890-3)

Further discussion

• Helge Maakestad, Notes on the Chern-character, Journal of Generalized Lie Theory and Applications, 2017, 11:1 (arXiv:math/0612060, doi:10.4172/1736-4337.100025)

• Goncalo Tabuada, A universal characterization of the Chern character maps (arXiv/1002.3276)

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

• Hilton 71, Theorem 3.18

• Yuli Rudyak, II.7.13 in: On Thom Spectra, Orientability, and Cobordism, Springer 1998 (doi:10.1007/978-3-540-77751-9)

• Jacob Gross, The homology of moduli stacks of complexes (arXiv:1907.03269)

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

• Victor Buchstaber, The Chern–Dold character in cobordisms. I,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (mathnet:3530)

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 (doi:10.1070/SM1970v012n04ABEH000939)

Review in

• Hilton 71, p. 50

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

• Hilton 71, Theorem 5.8.

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

• John Lind, Hisham Sati, Craig Westerland, Section 2.1: Twisted iterated algebraic K-theory and topological T-duality for sphere bundles, Ann. K-Th. 5 (2020) 1-42 (arXiv:1601.06285)

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):

• Mike Hopkins, Isadore Singer, Section 4.8, page 47 of Quadratic Functions in Geometry, Topology,and M-Theory, (math.AT/0211216).

• Ulrich Bunke, Differential cohomology (arXiv:1208.3961)

• Ulrich Bunke, David Gepner, around def. 2.1 of: Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory (arXiv:1306.0247)

• Markus Upmeier, Refinements of the Chern-Dold Character: Cocycle Additions in Differential Cohomology, J. Homotopy Relat. Struct. 11, 291–307 (2016). (arXiv:1404.2027, doi:10.1007/s40062-015-0106-y)

• Ulrich Bunke, Thomas Nikolaus, Twisted differential cohomology, Algebr. Geom. Topol. Volume 19, Number 4 (2019), 1631-1710. (arXiv:1406.3231, euclid:euclid.agt/1566439272)

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

• Victor Buchstaber, A. P. Veselov, Chern-Dold character in complex cobordisms and abelian varieties (arXiv:2007.05782)

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

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, section 4.4. of Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

based on Bunke-Gepner 13.

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalizaed, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]

The equivariant Chern-Dold character in equivariant cohomology:

• Wolfgang Lück, Chern characters for proper equivariant homology theories and applications to K- and L-theory, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 (doi:10.1515/crll.2002.015, pdf)

• Wolfgang Lück, Equivariant Cohomological Chern Characters, International Journal of Algebra and Computation, Vol. 15, No. 05n06, pp. 1025-1052 (2005) (arXiv:math/0401047, doi:10.1142/S0218196705002773)

• Wolfgang Lück, Equivariant Chern characters, 2006 (pdf, pdf)

In derived algebraic geometry

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

• Bertrand Toën, Gabriele Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry (arXiv:0804.1274)

which conjectures a construction that is fully developed in

• Bertrand Toën, Gabriele Vezzosi, Caractères de Chern, traces équivariantes et géométrie algébrique dérivée (arXiv:0903.3292)

See also

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

• Marc Hoyois, Sarah Scherotzke, Nicolò Sibilla, Higher traces, noncommutative motives, and the categorified Chern character (arXiv:1511.03589)