Let (i) $c\in B_r(M),\omega \in Z^r(M)$ or (ii) $\omega\in B^r(M),c\in Z_r(M).$ Show in both cases $$(c,\omega)=0.$$ My approach to prove (ii), $$(c,\omega)=\int_c\omega$$ if we use Stokes theorem $$\int_c \omega=\int_{\partial c} \psi $$ where we can let $\omega=d \psi$ because $\omega\in B^r(M)$. Since $c\in Z_r(M)$ we have $\partial c =\emptyset$ which proves $$(c,\omega)=0.$$
How do I prove this for the case given by (i).