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Can someone provide an explanation of the intuition behind De Rham Cohomology, and what exactly one is trying to achieve?

Assuming the audience knows of manifolds, tangent spaces, cotagent spaces, and differential forms, what is a proper and insightful reasoning to give of the set-up of De Rham Cohomology and how it came to be?

In essence, what is the elevator pitch? And if you have a moment to expand on said pitch, that would also be quite appreciated.

I know, for example, that there's some game going on with exact and closed forms that relates to the existence of holes in a manifold, but would be quite happy to hear a further explanation.

Anthony Peter
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    I have to imagine this has been asked many times on this website and others. See eg here. –  Nov 14 '17 at 19:33
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    What I'm usually saying is that it generalises the questions "Which curl-free vector fields are not gradients? Which divergence-free fields are not curl of something? And what does this have to do with topology?" But as Mike Miller said, a lot can be found elsewhere on mse. – Peter Franek Nov 14 '17 at 19:53
  • All of the fun integral tricks you do in complex analysis are examples of cohomology! –  Nov 14 '17 at 19:56

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