universal principal bundle

(see also *Chern-Weil theory*, parameterized homotopy theory)

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.

See at *classifying space*.

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)

Revised on February 9, 2017 05:06:33
by Urs Schreiber
(83.208.22.80)