nLab Cheeger-Simons homomorphism

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

• Jeff Cheeger, James Simons, Section 2 of: Differential characters and geometric invariants, pages 50–80 in: Geometry and Topology, Lecture Notes in Mathematics, vol 1167. Springer (1985) (pdf, doi:10.1007/BFb0075212)

• Michael Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)