nLab Cheeger-Simons homomorphism

Contents

Context

Differential cohomology

differential cohomology

Contents

Idea

Where the Chern-Weil homomorphism assigns characteristic forms in de Rham cohomology to principal bundles, the Cheeger-Simons homomorphism refines this to an assignment of secondary characteristic classes in ordinary differential cohomology to principal connections:

$\array{ G Connections(X)_{/\sim} &\overset{ \overset{ \color{blue} \text{Cheeger-Simons homomorphism} }{ cs_G } }{\longrightarrow}& Hom \Big( H^\bullet(B G;\, \mathbb{Z}) \,,\, \widehat H^\bullet(X) \Big) \\ \big\downarrow && \big\downarrow \\ G Bundles(X)_{/\sim} &\underset{ \underset{ \color{blue} \text{Chern-Weil homomorphism} }{cw_G} }{\longrightarrow}& Hom \Big( imv^\bullet(\mathfrak{g}) \,,\, H_{dR}^\bullet(X) \Big) }$

References

Created on September 14, 2020 at 12:42:01. See the history of this page for a list of all contributions to it.