Contents

Idea

For $G$ a Lie group with Lie algebra $𝔤$, a $G$-principal bundle $P\to X$ on a smooth manifold $X$ induces a collection of classes in the de Rham cohomology of $X$: the classes of the curvature characteristic forms

of the curvature 2-form $F_A\in {\Omega }^2(P,𝔤)$ of any connection on $P$, and for each invariant polynomial $⟨-⟩$ of arity $n$ on $𝔤$.

This is a map from the first nonabelian cohomology of $X$ with coefficients in $G$ to the de Rham cohomology of $X$

where $i$ runs over a set of generators of the invariant polynomials. This is the analogy in nonabelian differential cohomology of the generalized Chern character map in generalized Eilenberg-Steenrod-differential cohomology.

Refined Chern-Weil homomorphism

We describe the refined Chern-Weil homomorphism (which associates a class in ordinary differential cohomology to a principal bundle with connection) in terms of the universal connection on the universal principal bundle. We follow (HopkinsSinger, section 3.3).

• Let $G$ be a compact Lie group

• with Lie algebra $𝔤$;

• and write $\mathrm{inv}(𝔤)$ for the dg-algebra of invariant polynomials on $𝔤$ (which has trivial differential).

• Write $B^{(n)}G$ for the smooth level $n$ classifying space

• and $BG:={{\lim }_{\to }}_n{B}^{(n)}G$ for the colimit, a smooth model of the classifying space of $G$.

• Write ${\nabla }_{\mathrm{univ}}$ for the universal connection on $EG\to BG$.

• Let $[c]\in H^k(BG,\mathbb{Z})$ be a characteristic class

• and choose a refinement $[\hat{c}]\in H_{\mathrm{diff}}^k(BG)$ in ordinary differential cohomology represented by a differential function

  $$(c,h,w)\in C^k(BG,\mathbb{Z})\times C^{k-1}(BG,ℝ)\times (W(𝔤)\simeq C^k(BG,ℝ){)}^k.$$(c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times (W(\mathfrak{g}) \simeq C^k(B G, \mathbb{R}))^k \,.

Examples

• Gauss-Bonnet theorem

(…)

References

A classical textbook reference is

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

The description of the refined Chern-Weil homomorphism in terms of differential function complexes is in section 3.3. of

• Mike Hopkins, I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory

For more references see Chern-Weil theory.