nLab Milnor construction

Idea

Given a (Hausdorff) compact 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 G-CW-complex. 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 bundle $G$-principal bundles over all Hausdorff topological spaces.

There is a generalisation of Milnor’s construction that works for topological groupoids, and reduces to the above case when the groupoid is $\mathbf{B}G$, the delooping of the group $G$.

Related entries

• classifying space

• universal principal bundle

• bar construction

References

• John Milnor, Construction of Universal Bundles, I, Ann. of Math. 63:2 (1956) 272-284 jstor; Construction of Universal Bundles, II, Ann. of Math. 63:3 (1956) 430-436, jstor; reprinted in Collected Works of John Milnor, gBooks.

• Wikipedia: Classifying Space.

• John Milnor, James Stasheff, Characteristic Classes, Princeton University Press.

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, The Milnor Construction: Homotopy Classification of Principal Bundles, doi, in:

  Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Vol. 726 (2008) 75-81.