General-group characteristic refinement data

Assume full AC, fix an integer $k\ge 1$, and let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold without boundary. Let $G$ be a finite-dimensional real Lie group with Lie algebra $\mathfrak{g}$, and choose a classifying space $BG$ with universal principal $G$-bundle $EG\to BG$. A smooth principal $G$-bundle $P\to M$ is a smooth right $G$-action on $P$ for which $P\to M$ is locally equivariantly isomorphic to $U\times G\to U$; a classifying map $f_P:M\to BG$ classifies its underlying bundle, and $c(P):=f_P^{\ast }c$ for $c\in H^k(BG;\mathbb{Z})$. A principal connection is a $\mathfrak{g}$-valued one-form $A\in {\mathrm{\Omega }}^1(P;\mathfrak{g})$ with $R_g^{\ast }A={\mathrm{Ad}}_{g^{-1}}A$ and $A({\xi }_P)=\xi$ for every $g\in G$, $\xi \in \mathfrak{g}$, and fundamental vector field ${\xi }_P$. Its curvature is $F_A=dA+\frac{1}{2}[A\wedge A]$.

For even $k=2m$, choose an $\mathrm{Ad}(G)$-invariant symmetric polynomial $p\in (S^m{\mathfrak{g}}^{\ast }{)}^G$ with a fixed real normalization, including all $2\pi i$ factors. Its evaluation $p(F_A,\dots ,F_A)$ is a scalar form on $P$; require it to be basic (horizontal and right-$G$-invariant), and define $\omega \in {\mathrm{\Omega }}^{2m}(M)$ as its unique descended form. For odd $k$, set $\omega =0$. Require $\omega$ to be closed and $[\omega {]}_{\mathrm{d}\mathrm{R}}=c(P{)}_{\mathbb{R}}$. A general-group characteristic refinement datum additionally specifies a degree-$k$ differential character $\widehat{c}(P,A)\in {\widehat{H}}^k(M;\mathbb{Z})$ with $I(\widehat{c})=c(P)$ and $R(\widehat{c})=\omega$. This is supplied compatible data, not an existence or uniqueness assertion for arbitrary $G$.

If $c$ and $p$ are pulled back from a unitary representation $G\to U(r)$ and its normalized Chern–Weil polynomial, the unitary-group result applies to that pulled-back pair. For fixed compatible $c(P)$ and $\omega$, any two supplied refinements differ by an element of $\ker I\cap \ker R\cong H_{\mathrm{d}\mathrm{R}}^{k-1}(M;\mathbb{R})\mathrm{/}\mathrm{im}(H^{k-1}(M;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^{k-1}(M;\mathbb{R}))\mathrm{.}$ The torsion part of $c(P)$ is invisible to real curvature; any desired flat torsion refinement must be separately specified.

Cheeger–Simons differential character

Exact sequences for ordinary differential cohomology

Characteristic forms represent topological characteristic classes over the reals

The image $c(P{)}_{\mathbb{R}}$ is induced by $\mathbb{Z}\hookrightarrow \mathbb{R}$, and $p(F_A,\dots ,F_A)$ means evaluation of the symmetric $m$-linear polynomial on the curvature $m$ times. If $p(F_A,\dots ,F_A)$ fails to be basic, closed, or compatible with $c(P{)}_{\mathbb{R}}$, no datum of the stated type has been supplied. Compatibility is an explicit hypothesis; the cited Chern–Weil suppliers cover their own stated classical vector-bundle scope and do not establish arbitrary-$G$ existence. The ambiguity among supplied refinements with fixed $I$ and $R$ is described in Ambiguity of a differential refinement with fixed characteristic class and curvature.

The Chern–Weil compatibility supplier Characteristic forms represent topological characteristic classes over the reals covers its stated classical characteristic forms, not arbitrary invariant polynomials or arbitrary integral classes. Compatibility is therefore an explicit hypothesis of this interface. No arbitrary-$G$ existence theorem is asserted.