universal connection (changes) in nLab

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Contents

Idea

For GG a compact Lie group there is a way to equip the topological classifying space BGB G with smooth structure such that the corresponding smooth universal principal bundle EG→BGE G \to B G carries a smooth connection ∇ univ\nabla_{univ} with the property that for every GG-principal bundle P→XP \to X with connection ∇\nabla there is a smooth representative f:X→BGf : X \to B G of the classifying map, such that (P,∇)≃(P,f *∇ univ)(P, \nabla) \simeq (P, f^* \nabla_{univ}). This ∇ univ\nabla_{univ} is called the universal GG-connection.

• connection on a bundle

• universal principal ∞-bundle , groupal model for universal principal ∞-bundles

• universal connection

References

• M. S. Narasimhan and S. Ramanan,

  Existence of universal connections , Amer. J. Math. 83 (1961), 563–572. MR 24 #A3597

  Existence of universal connections II , Amer. J. Math. 85 (1963), 223–231. MR 27 #1904

In bounded dimension

Universal connections for manifolds of some bounded dimension ≤n\leq n are appealed to in

• Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

  (in the context of Chern-Weil theory)

and discussed in detail in

• Mudumbai Narasimhan and Sundararaman Ramanan, Existence of Universal Connections, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 (jstor:2372896)

• Mudumbai Narasimhan and Sundararaman Ramanan, Existence of Universal Connections II, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 (jstor:2373211)

• Roger Schlafly, Universal connections, Invent Math 59, 59–65 (1980) (doi:10.1007/BF01390314)

• Roger Schlafly, Universal connections: the local problem, Pacific J. Math. Volume 98, Number 1 (1982), 157-171 (euclid:pjm/1102734394)

See also

• Shrawan Kumar, A Remark on Universal Connections, Mathematische Annalen 260 (1982): 453-462 (dml:163680)

In unbounded dimension

Discussion of universal connections on some smooth incarnation of the full classifying space:

Using diffeological spaces:

• Mark Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 (euclid:jdg/1214434974)