Our goal is to interpreting a formula: $$ \int_{\Sigma_d} a_d =\int_{M_D} a_d \cup PD(\Sigma_d) $$ where $a_d$ is a $d$-form on $\Sigma_d$ and $PD$ is the Poincare dual to the $\Sigma_d$. The $\Sigma_d$ is the submanifold of $M_D$, with $D>d$.
Is this formula only true under certain conditions? Like $a_d$ is closed?
A simple proof for this formula.