nLab universal connection

Idea

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

Related concepts

• connection on a bundle

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

• universal connection

References

In bounded dimension

Universal connections for manifolds of some bounded dimension $\leq n$ 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)