Let $M$ be smooth oriented manifold with boundary $\partial M $, dim$M=n$. The two short exact sequences in de Rham cohomology and singular homology
$$0\longrightarrow{}\Omega^{}(M, \partial M)\xrightarrow{\ j_{} \ }\Omega^{}(M)\xrightarrow{\ i^{} \ }\Omega^{*}(\partial M )\xrightarrow{\ \ }0$$ and $$0\longrightarrow{}C_{}(\partial M )\xrightarrow{\ i_{} \ }C_{}(M)\xrightarrow{\ j_{} \ }C_{*}(M, \partial M)\xrightarrow{\ \ }0$$ induce these long exact sequence in cohomology de Rham and singular homology $$\cdots \longrightarrow{}H^{n-k}_{dR}(M)\xrightarrow{\ i^{*} \ }H^{n-k}_{dR}(\partial M )\xrightarrow{\ \delta \ }H^{k+1}_{dR}(M,\partial M)\xrightarrow{\ j_{*} \ }H^{k+1}_{dR}(M)\longrightarrow\cdots$$\
$$\cdots \longrightarrow{}H_{k}(M, \partial M)\xrightarrow{\ \partial \ }H_{k-1}(\partial M)\xrightarrow{\ i_{*} \ }H_{k-1}({M})\xrightarrow{\ j_{*} \ }H_{k-1}(M, \partial M)\longrightarrow\cdots$$
We define the the map $I_{M} : H^{n-k}(M, \partial M)\rightarrow H_{k}(M), \ \ [M]\frown\omega$, where [M] fundamental class M.
$I_{\partial M} : H^{n-k}(\partial M)\rightarrow H_{k}(\partial M), \ \ \partial[M]\frown\omega$, where $\partial [M]$ fundamental class $\partial M$.
How to prove that $ i_{*}\circ I_{\partial M}$ and $I_{M}\circ \delta$ commute up to signe $(-1)^{n-k+1}$ ?