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.