Universal characteristic class - Imported Knowlpedia Content

Definition

Let $G$ be a topological group, let $R$ be a coefficient ring, and choose a classifying space $BG$ . A universal characteristic class of degree $n$ is a class

c\in H^n(BG;R).

For a principal $G$ -bundle $P\to B$ over a paracompact base and a classifying map $f_P:B\to BG$ , its associated characteristic class is

c(P):=f_P^*c\in H^n(B;R).

Because any two classifying maps for $P$ are homotopic, $c(P)$ is independent of the choice of $f_P$ .

Naturality and universality

If $g:B'\to B$ , then

c(g^*P)=g^*c(P),

so a universal class determines a natural characteristic class of principal $G$ -bundles. Conversely, in the standard homotopy-theoretic setting, every such cohomological natural assignment is obtained by evaluating it on the universal bundle $EG\to BG$ . This representability principle is developed for characteristic classes in Milnor and Stasheff, Chapter 4.

Changing the model of $BG$ transports $c$ through a homotopy equivalence and therefore does not change the resulting assignment.

Standard examples

The universal Stiefel–Whitney classes lie in $H^i(BO(n);\mathbb Z/2)$ , the universal Chern classes in $H^{2i}(BU(n);\mathbb Z)$ , and the universal Pontryagin classes in $H^{4i}(BO(n);\mathbb Z)$ . Pulling them back along classifying maps gives the corresponding classes of real or complex vector bundles through their frame bundles.

The phrase “universal Chern class” names one of these specific universal classes; it is an example, not a synonym for an arbitrary class in $H^*(BG;R)$ .

Scope and conventions

The definition depends on a cohomology theory and coefficients. This knowl uses ordinary singular cohomology, but generalized cohomology theories give analogous universal classes in $h^n(BG)$ . A class for $G$ -bundles can also be transported along a homomorphism $H\to G$ by the induced map $BH\to BG$ .

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974. DOI record. Relevant: Chapter 4, universal bundles and characteristic classes.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 14, characteristic classes and universal constructions.