I encounter the pairing between cycle and cocycle a lot, but I don't really understand it.
So the basic setiing looks like this: We have a manifold $M$, we have $[\omega]\in H^k(M;R)$ and $[A]\in H_k(M;R)$. Here I'm only considering the coefficient ring $R$ to be $\mathbb{R}$ or $\mathbb{Z}$. $$H^k(M;R) \times H_k(M;R) \rightarrow R$$
The above pairing is just the natural evaluataion map of a linear functional.
I was wondering the following: Is this pairing the same as integration in the de rham setting? i.e. do we have $\langle[\omega],[A]\rangle=\int_A \omega$? If so, how does one see this and what are the conditions we need for this to be true?