nLab
Chern-Dold character

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Rational homotopy theory

Contents

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 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).

Exanples

Examples of Chern-Dold characters:

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

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:

Created on October 4, 2020 at 07:52:57. See the history of this page for a list of all contributions to it.