bundles

# Contents

## Idea

Given a (Hausdorff) topological group $G$, the Milnor construction of the universal principal bundle for $G$ (also known as Milnor’s join construction) constructs the join of infinitely many copies of $G$, i.e., the colimit of joins

$(E G)_{Milnor} := colim \; G \ast G \ast \ldots \ast G,$

and canonically equips it with a continuous and free right action of $G$ that yields the structure of a CW-complex such that the action of $G$ permutes the cells. Consequently, the natural projection $(E G)_{Milnor} \to (E G)_{Milnor}/G$ is a model for the universal bundle $E G \to B G$ of locally trivial principal $G$-bundles over paracompact Hausdorff spaces, or equivalently, of numerable principal $G$-bundles over all Hausdorff topological spaces.

## References

Revised on November 20, 2015 12:04:39 by Urs Schreiber (86.187.57.85)