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Given a principal $G$-bundle $P \to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism $$S^*(g)^G \to H_{DR}^*(M)$$ associated to any invariant polynomial on $g$, a characteristic class in the de Rham cohomology of $M$. If $G$ is compact and the bundle is the universal bundle $E_G \to B_G$, then the Chern-Weil homomorphism can be shown to be an isomorphism.

So far I have seen in the literature only the construction of the homomorphism for smooth finite dimensional principal bundles, and one construction and proof of the isomorphism in the case of classifying spaces of compact Lie groups in Dupont's book "Curvature and Characteristic Classes". In this book, the author introduces the notion of simplicial manifolds and their de Rham cohomology, in order to work through the fact that $B_G$ is not a smooth finite dimensional manifold, and therefore differential forms are not well-defined a priori.

My question is the following: are there any more direct constructions of the Chern-Weil isomorphism in the case where $M = B_G$ ?

I am thinking of the following: we could filtrate $E_G \to B_G$ by smooth finite dimensional principal bundle $E_jG \to B_j G$ on which $G$ acts freely, so that we get a homomorphism at each stage $j$, and then conclude at the limit, defining a differential form on $B_G$ as a class in the limit $\underset{j}{\lim} \Omega^*(B_j G)$. It seems to work for the universal bundle $S^{\infty} \to \mathbb{C}^{\infty}$ for $G = S^1$, but I might be missing something.

Thanks a lot

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  • $\begingroup$ Did you get to see any proof? $\endgroup$ – Praphulla Koushik Dec 26 '18 at 2:56

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