## Simply-Connected Spaces and Cohomology

We’ve seen that if a manifold is simply-connected then the first degree of cubic singular homology is trivial. I say that the same is true of the first degree of de Rham 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

for any closed curve . But this means that every closed -form is path-independent, and path-independent -forms are exact. And so we conclude that , as asserted.

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.