Simply-Connected Spaces and Cohomology
Indeed, say that is simply-connected, so that any closed curve can be written as the boundary of some surface . Then we take any closed -form with . Stokes’ theorem tells us that
It will (eventually) turn out that the fact that both and vanish together is not a coincidence, but is in fact an example of a much deeper correspondence between homology and cohomology — between topology and analysis.
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