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 infinite join of copies of $G$, i.e. the colimit of joins

$$(EG{)}_{\mathrm{Milnor}}:=\mathrm{colim}G*G*\dots *G$$(E G)_{Milnor} := colim G \ast G \ast \ldots \ast G

and canonically equipps with a continuous free right action of $G$ which permits a structure of a CW-complex such that the action of $G$ permutes its cells. Consequently, the natural projection $(EG{)}_{\mathrm{Milnor}}\to (EG{)}_{\mathrm{Milnor}}/G$ is a model for the universal bundle $EG\to BG$ 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, 272-284 (1956) jstor; II, Ann. of Math. 63:3 (1956) 430-436, jstor; reprinted in Collected Works of John Milnor, gBooks

• classifying space, universal principal bundle

• wikipedia: classifying space

• John W. Milnor, James Stasheff, Characteristic classes, Princeton Univ. 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, 2008, vol. 726 (2008) 75-81