nLab universal principal bundle

Idea

For $G$ a topological group there is a notion of $G$-principal bundles $P \to X$ over any topological space $X$. Under continuous maps $f : X \to Y$ there is a notion of pullback of principal bundles $f^* : G Bund(Y) \to G Bund(X)$.

A universal $G$-principal bundle is a $G$-principal bundle, which is usually written $E G \to B G$, such that for every CW-complex $X$ the map

$$[X, B G] \to G Bund(X)/_\sim$$

from homotopy classes of continuous functions $X \to B G$ given by $[f] \mapsto f^* E G$, is an isomorphism.

In this case one calls $B G$ a classifying space for $G$-principal bundles.

The universal principal bundle is characterized, up to equivalence, by its total space $E G$ being contractible.

More generally, we can ask for a universal bundle for numerable bundles, that is principal bundles which admit a trivialisation over a numerable open cover. Such a bundle exists, and classifies numerable bundles over all topological spaces, not just paracompact spaces or CW-complexes.

Details

See at classifying space.

Related concepts

• classifying space

• Milnor construction

• universal vector bundle, universal complex line bundle

• universal principal infinity-bundle.

References

General

Among the earliest references that consider the notion of universal bundles is

• Shiing-shen Chern, Sun, … (1949)

A review is for instance in

• Stephen Mitchell, Universal principal bundles and classifying spaces (pdf)
• Dale Husemoeller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) gBooks pdf

For more see the references at classifying space.

For equivariant bundles

Discussion of universal equivariant principal bundles:

• Tammo tom Dieck, Faserbündel mit Gruppenoperation, Arch. Math 20, 136–143 (1969) (doi:10.1007/BF01899003)

• Richard Lashof, Equivariant bundles, Illinois J. Math. 26(2): 257-271, 1982 (doi:10.1215/ijm/1256046796.full)

• Peter May, Some remarks on equivariant bundles and classifying spaces, Théorie de l’homotopie, Astérisque, no. 191 (1990), 15 p. (numdam:AST_1990__191__239_0)

• Mitutaka Murayama, Kazuhisa Shimakawa, Universal equivariant bundles, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 (doi:10.1090/S0002-9939-1995-1231040-9)

• Bernardo Uribe, Wolfgang Lück, Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 (arXiv:1304.4862, doi:10.2140/agt.2014.14.1925)

• Bertrand Guillou, Peter May, Mona Merling Categorical models for equivariant classifying spaces, Algebr. Geom. Topol. 17 (2017) 2565-2602 (arXiv:1201.5178, doi:10.2140/agt.2017.17.2565)