Milnor construction

Given a (Hausdorff) topological group GG, the Milnor construction of universal principal GG-bundles (also known as the Milnor’s join construction) constructs the infinite join of copies of GG, i.e. the colimit of joins

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

and canonically equipps with a continuous free right action of GG which permits a structure of a CW-complex such that the action of GG permutes its 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 EGBGE G\to B G of locally trivial principal GG-bundles over paracompact Hausdorff spaces, or equivalently, of numerable principal GG-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

Created on May 18, 2012 21:47:27 by Zoran Škoda (