nLab universal connection

Contents

Context

Differential cohomology

differential cohomology

Contents

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.

References

In bounded dimension

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

and discussed in detail in

• 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)

Last revised on August 28, 2020 at 11:20:01. See the history of this page for a list of all contributions to it.