Characteristic class of principal bundle

Asked Feb 27 '22 at 12:19

Active Feb 27 '22 at 13:42

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I've been told that principal $G$-bundles $E \to M$ are classified by specifying a characteristic class $c(E) \in H^2(M,\pi_1(G)) ≈ \pi_1(G)$

I have a few questions

Given a bundle $E$ how do we get a $c(E)$ and conversely?

I've seen something like this before where the Picard group is isomorphic to $H^1(O^*)$ because of transition functions but I'm not sure why here we have the second cohomology and the sheaf is the constant sheaf of fundamental group?

Why is the second sheaf cohomology isomorphic to $\pi_1(G)$?

Here M is a smooth orientable surface and G is a Lie group

asked Feb 27 '22 at 12:19

Angry_Math_Person

2,678

Do you know about classifying spaces? Have you seen this question? – Michael Albanese Feb 27 '22 at 13:35
Oh wow. Seems like someone ran into the same problem. I'll read the answer in that post. Thanks. – Angry_Math_Person Feb 27 '22 at 13:40
Flag it as duplicate? – C.F.G Mar 01 '22 at 18:19

Characteristic class of principal bundle

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