\begin{tikzcd}[column sep=small, row sep=small]
 & & \mathring{Q} \arrow[d] \\
 & W & Q \arrow[l, hook'] \\
\mathring{P} \arrow[r, hook] & P \arrow[u, hook] & P \cap Q = R = \partial P \arrow[l, hook'] \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & H^{i-1}(R, F) \arrow[dl, no head] \arrow[dr] \arrow[d] & \\
H^{n-i}(P, \check{F} \otimes \mathcal{T}_W)^\vee \simeq H^{i}_{!}(\mathring{P}, F) \arrow[r] & H^{i}(W, F) \arrow[dl] \arrow[dr] & H^{i}_{!}(\mathring{Q}, F) \simeq H^{n-i}(Q, \check{F} \otimes \mathcal{T}_W)^\vee \arrow[l] \\
H^{i}(P, F) \arrow[r] & H^{i}(R, F) & H^{i}(Q, F) \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & T = \partial M & & t \arrow[ll] \\
 & W & Q \arrow[l, hook'] \arrow[u, "2"'] & & N = D_{1} \sim e \arrow[ll, hook'] \arrow[u, no head] \\
\partial N = S^{1} & P \arrow[l] \arrow[u, leftrightarrow] & S^{1} \times T \arrow[l, hook'] \arrow[u, hook] & & \partial N = S^{1} \arrow[ll, leftrightarrow] \arrow[u, hook] \\
s \arrow[u] & M \arrow[l] \arrow[u, hook] & \partial M = T \arrow[l, hook'] \arrow[u, leftrightarrow] & & (s, t) \arrow[ll] \arrow[u, hook]
\end{tikzcd}

\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & H^{i-1}(R) \arrow[d] & & \\
H^{n-i}(P)^\vee \arrow[r] & H^{i}(W) \arrow[dl] \arrow[dr] & H^{n-i}(Q)^\vee \arrow[l] & \\
H^{i}(P) \arrow[r] & H^{i}(R) \arrow[d] & H^{i}(Q) \arrow[l] \arrow[r] & H^{n-i-1}(P)^\vee \\
 & H^{i+1}(W) & &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & L^{\bullet\,\mathrm{triv}} = \mathbb{R}\Gamma(W_0)^{\mathrm{triv}} \arrow[dl, dashed] & \\
{[\mathbb{R}^\bullet]} \simeq \mathbb{R}(\Gamma, \pi)_{!}(\mathring{M}) \arrow[rr, dashed] & & K^\bullet = \mathbb{R}(\Gamma, \pi)(M) \arrow[ul, "\rho"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & L^{\bullet\pi} \arrow[dl, dashed] & \\
{[\mathbb{R}^{\bullet\pi}]} = \mathbb{R}\Gamma_{!}(\mathring{P}) \arrow[rr, dashed] & & K^{\bullet\pi} = \mathbb{R}\Gamma(P) = \mathbb{R}\Gamma(W - W_0) \arrow[ul, "{(\alpha,\ \beta)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & L^\bullet = \mathbb{R}\Gamma(W_0) \arrow[dl, dashed] & \\
{[\mathbb{R}^{\omega}]} \simeq \mathbb{R}\Gamma_{!}(\mathring{M}) \arrow[rr, dashed] & & K^{\bullet\omega} = \mathbb{R}\Gamma(M) \arrow[ul, "\text{compatible avec cup}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(P) \simeq \mathbb{R}\Gamma(W - W_0) \arrow[dl, "\beta"'] & \\
\mathbb{R}\Gamma(W_0)[-2] \arrow[rr] & & {[\mathbb{R}\Gamma_{!}(W)]} \arrow[ul, "\text{compatible avec cup}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(W_0) \simeq \mathbb{R}\Gamma(Q) \simeq \mathbb{R}\Gamma(\mathring{Q}) \arrow[dl, dashed, "\beta'"'] \\
\mathbb{R}\Gamma_{!}(\mathring{P}) \arrow[r, dashed] & \mathbb{R}\Gamma(W) \arrow[u, "\rho_1"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & L^{\bullet\,\mathrm{triv}} \arrow[dl, "\rho'"'] & \\
\mathbb{R}\mathrm{Hom}_{\Lambda}(K^\bullet, \Lambda)[-2n] \arrow[rr] & & K^\bullet \arrow[ul, "\rho"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & L^\bullet \simeq \mathbb{R}\Gamma(W_0) \arrow[dl] & \\
\mathbb{R}\mathrm{Hom}_{\Lambda}(K^{\bullet\,\mathrm{ens}}, \Lambda)[-2n] \arrow[rr] & & K^{\bullet\,\mathrm{ens}} \simeq \mathbb{R}\Gamma(M) \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & L^\bullet \overset{\mathbf{L}}{\otimes}_{\Lambda} \mathbb{R}\Gamma(S^1) \simeq L^\bullet \times L^\bullet[-1] \arrow[dl, "{(\alpha',\ \beta')}"'] & \\
\mathbb{R}\mathrm{Hom}_{\Lambda}(K^{\bullet\,\pi}, \Lambda)[-2n-1] \arrow[rr] & & K^{\bullet\,\pi} \simeq \mathbb{R}\Gamma(P) \simeq \mathbb{R}\Gamma(W - W_0) \arrow[ul, "{(\alpha,\ \beta)}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small]
H^{n-1}(M) \arrow[r] & H^{n-1}(W_0) \arrow[r, "\mathrm{surj}"] & H^{n}(M)^\vee \arrow[d] \\
H^{n-1}(M)^\vee & H^{n}(W_0) \arrow[l] & H^{n}(M) \arrow[l]
\end{tikzcd}