Milnor construction



Given a (Hausdorff) 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) Milnor:=colimG*G**G, (E G)_{Milnor} := colim \; G \ast G \ast \ldots \ast G,

and canonically equips it with a continuous and free right action of G G that yields the structure of a CW-complex such that the action of G G permutes the cells. Consequently, the natural projection (EG) Milnor(EG) Milnor/G(E G)_{Milnor} \to (E G)_{Milnor}/G is a model for the universal 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.


Revised on May 26, 2017 05:38:27 by Urs Schreiber (