Do periods of differential forms only depend on their cohomology class?

Question

The periods of a closed differential form are the values of the integration of the form along integral homology cycles. Is it correct that the periods depend only on the cohomology class i.e. do two closed forms in same cohomology class have same periods?

Answer 1

Yes. In general, the integral of a differential form $\omega$ over a closed manifold (i.e. without boundary) only depends on the cohomology class of $\omega$. This follows immediately from Stokes' theorem: if $d\eta = \omega - \omega'$, then

$$\int_M \omega-\omega' = \int_M d\eta = \int_{\partial M} \eta = \int_\emptyset \eta = 0.$$

Since periods are integrals over closed cycles, they are really invariants of cohomology classes.