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

$$(E G)_{Milnor} \;\coloneqq\; \underset {\underset{n \in \mathbb{N}}{\longrightarrow}} {\lim} \; \big( \underset {n \; factors} {\underbrace{G \ast G \ast \ldots \ast G}} \big)$$

and canonically equips it with a continuous and free right group action of $G$ that yields the structure of a G-CW-complex. Consequently, the natural quotient space projection $(E G)_{Milnor} \to (E G)_{Milnor}/G$ is a model for the universal principal 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:1969609]

• John Milnor, Construction of Universal Bundles II, Ann. of Math. 63 3 (1956) 430-436 [jstor:1970012]

  reprinted in: Collected Works of John Milnor [gBooks]

• 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.