Let $G$ be a Lie group.

In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :

The notion of a topological principal $G$-bundle $\pi:E\rightarrow X$ on a topological space $X$ is defined exactly as in Definition $3.1$ (the definition of usual principal bundle over manifold), only the words "differentiable" and "diffeomorphism" are replaced by "continuous" and "homomorphism". The purpose of this and the following section is to show that the Chern-Weil morphism defines characteristic classes of topological G-bundles.

But does not mention (or I could not find) anything about characteristic classes of topological principal bundles. Are there other places that discuss the notion of characteristic classe of topological principal bundles using Chern-Weil theory?

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    $\begingroup$ At the risk of saying something stupid, Chern-Weil theory involves the notions of connections and curvature, so requires at least a $C^{2}$-regularity on your bundle and manifold for their naive definition. If you want to relax this regularity, you're back in the world of classifying spaces and their cohomology to define characteristic classes in this language. I'm sure you could work much harder to make headway in either direction, but already you don't define what you mean by topological, so doing so would be a good start. $\endgroup$ – Andy Sanders Jan 18 '20 at 17:39
  • $\begingroup$ I have mentioned what does a topological principal bundle mean to the author. $\endgroup$ – Praphulla Koushik Jan 18 '20 at 17:46
  • $\begingroup$ fair enough, I apologize for asking this: but my comment still stands. $\endgroup$ – Andy Sanders Jan 18 '20 at 17:47
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    $\begingroup$ I think we're basically in agreement, Chern-Weil theory for bundles that are not $C^{2}$ over a $C^{2}$-manifold are at most a niche area, and at worst, not sensibly defined. This doesn't mean that such an investigation isn't important, it just means that it goes from being "standard material" to a "research area." $\endgroup$ – Andy Sanders Jan 18 '20 at 17:50
  • $\begingroup$ I feel the same... He does discuss Chern Weil map for Lie groups, $I(G)\rightarrow H^*(BG,\mathbb{R})$.. He use the notion of simplicial manifold associated to the topological space $BG$ to do that.. Thus, he is able to write Chern Weil map for topological principal bundles $EG\rightarrow BG$, but still it does not explain how one would do for an arbitrary topological principal bundle $E\rightarrow X$ $\endgroup$ – Praphulla Koushik Jan 18 '20 at 17:55

I think "Chern–Weil theory" here is being used in a slightly nonstandard way. The main result of the chapter, Theorem 5.5, is really just a topological statement that elements of $H^*(BG,\Lambda)$ correspond exactly to characteristic classes (Definition 5.1), which are defined to be functorial associations of ($\Lambda$-valued) singular cohomology classes to isomorphism classes of topological bundles. This is combined with the calculation that the cohomology of $BG$ is calculated to be a certain invariant polynomial ring (assuming, eg, $G$ is compact), i.e. the Chern–Weil homomorphism, since the group $G$ is still a Lie group.

In traditional Chern–Weil theory (for smooth bundles on manifolds), one also has the isomorphism of de Rham with singular cohomology, and, most importantly, that the invariant polynomials can be evaluated on curvature forms to give de Rham classes.

But in the topological case at hand, one still has invariant polynomials giving rise to characteristic classes, which is what I think Dupont means here.

  • $\begingroup$ Thanks for the answer. Yes, It looks more like a non standard term.. I will wait for some more time to see if any one has anything else to say and then accept this answer.. $\endgroup$ – Praphulla Koushik Jan 26 '20 at 3:22

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