# nLab Milnor construction

Given a (Hausdorff) topological group $G$, the Milnor construction of universal principal $G$-bundles (also known as the 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.

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

• D. Husemöller, M. Joachim, B. Jurčo, M. 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.

Revised on September 7, 2014 22:37:08 by Leonard? (24.124.53.243)