nLab
Chern-Dold character

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

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: