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Cheeger-Simons homomorphism

Contents

Context

\infty-Chern-Weil theory

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:

GConnections(X) / cs GCheeger-Simons homomorphism Hom(H (BG;),H^ (X)) GBundles(X) / cw GChern-Weil homomorphism Hom(imv (𝔤),H dR (X)) \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.