This is related to Sec 5, Chpt 1 of Bott Tu Differential Forms in Algebraic Topology.
Given $S\subset M$ with $M$ a oriented manifold and $S$ a closed oriented submanifold, one takes $\omega\in Z_c^k(M)$ where $Z_c^k$ denotes $k-$th de Rham cocycle defined by $Ker(\Omega^k_c(M)\xrightarrow{d}\Omega^{k+1}_c(M))$ where $\Omega_c^\star$ are compactly supported differential forms. Now $i:S\to M$ as the inclusion map gives rise to $i^\star\omega$. Since $S$ is closed, $supp(i^\star(\omega))\subset S$ is closed and compact by $supp(i^\star(\omega))\subset supp(\omega)$ and $S$ closed. Hence $\int_S i^\star:Z_c^k(M)\to R$ is defined.
$\textbf{Q:}$ Then the book says by Stokes Thm, $\int_Si^\star\in Hom(H_c^k(M),R)$. There is no good reason to say $\int_Si^\star$ descends to cohomology map to $R$ unless Stokes Thm guarantees $\partial S=\emptyset$. Then in this case, it does descend to cohomology level map. Do I have to assume $\partial S=\emptyset$ here? If I assume simplicial cohomology and simplicial homology here, I do have $\partial S=\emptyset$ by standard algebraic topology version Poincare duality via cap product.