universal connection



\infty-Chern-Weil theory

Differential cohomology



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 EGBGE G \to B G carries a smooth connection univ\nabla_{univ} with the property that for every GG-principal bundle PXP \to X with connection \nabla there is a smooth representative f:XBGf : 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.


In bounded dimension

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

and discussed in detail in

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)

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