Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem:
\begin{eqnarray} \int_{\sigma} d\omega &=& \int_{\partial \sigma} \omega \end{eqnarray}
I found in several books that this integral is zero, they argue it's because the sphere is compact so its boundary is zero.
My question is how can I relate this with chains??, I menan, how is the chain for a n-sphere and why its boundary is zero...