nLab Chern-Dold character

Idea

The Chern-Dold character is the natural generalization of the Chern character from topological K-theory to any generalized (Eilenberg-Steenrod) cohomology theory. It is given essentially by rationalization of coefficient spectra.

Definition

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).

Exanples

Examples of Chern-Dold characters:

• Chern character on KU;

• Pontrjagin character on KO.

Further examples listed in FSS 20

References

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, Generalized, 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)