nLab Milnor construction







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

(EG) Milnorlimn(G*G**Gnfactors) (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 G that yields the structure of a G-CW-complex. Consequently, the natural quotient space projection (EG) Milnor(EG) Milnor/G(E G)_{Milnor} \to (E G)_{Milnor}/G is a model for the universal principal bundle EGBG E G \to B G of locally trivial principal G G -bundles over paracompact Hausdorff spaces, or equivalently, of numerable bundle GG-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 BG\mathbf{B}G, the delooping of the group GG.


Last revised on April 30, 2023 at 07:54:31. See the history of this page for a list of all contributions to it.