\[
\begin{array}{ccc}
V_1 & V_2 & V_3 \\
\Gamma''_1 & \Gamma''_2 & \Gamma''_3 \\
\Vert & \Vert & \Vert \\
\Gamma'_1 - \partial & \Gamma'_2 - \partial & \Gamma'_3 - \partial
\end{array}
\qquad \partial = \{x, y\}
\]

\[
\begin{aligned}
\Gamma_1 &= \Gamma_2 \cup \Gamma_3 = \Gamma'_2 \cup \Gamma'_3 = \Gamma''_1 \cup \Gamma''_2 \cup \partial \\
\Gamma_2 &= \Gamma_3 \cup \Gamma_1 = \Gamma'_3 \cup \Gamma'_1 = \Gamma''_3 \cup \Gamma''_1 \cup \partial \\
\Gamma_3 &= \Gamma_1 \cup \Gamma_2 = \Gamma'_1 \cup \Gamma'_2 = \Gamma''_1 \cup \Gamma''_2 \cup \partial
\end{aligned}
\]

\[
\Gamma^{*}_3 = \Gamma''_1 \cup \Gamma''_2 \cup \partial
\quad\Big|\quad
U_3 = V_1 \cup V_2 \cup \Gamma''_3
\quad\Big|\quad
V_3
\]

\[
\mathcal{X} \simeq \overline{W} \amalg_{(\coprod_i B_i)} \Bigl(\coprod_i \overline{U_i}\Bigr)
\simeq \coprod_{i\,(W)} X_i \qquad X_i \simeq \mathcal{X}_i
\]

\[
\begin{array}{c}
e \in E \\
\cup \\
M(e) = M(E_e) \subset E_0 = M(E) \\
\big\downarrow{\scriptstyle \varphi} \\
I \ \text{ordonné}
\end{array}
\qquad
\begin{cases}
|E| = X \supset |E_e| \\
X_i = \bigcup_{\varphi(M(e)) \leq i} |E_e|
\end{cases}
\qquad
E_e = \{x \in E \mid x \leq e\}
\]

\[
\begin{array}{ccccccc}
\Sigma_{\{i_0, \dots, i_{k-1}\}}(\mathcal{X}) & \supset & \partial\Sigma_{\{i_0, \dots, i_{k-1}\}} & \supset & \Delta_{i_0 \dots i_k}(\mathcal{X}) & \supset & \partial\Delta_{i_0 \dots i_k} \\
\big\downarrow & & \big\downarrow & & \big\downarrow & & \big\downarrow \\
\Sigma_{i_{k-1}}(X)_{\ill{}} & & & & & & \\
\big\downarrow & & & & & & \\
X_{i_{k-1}} & \supset & \partial X_{i_{k-1}} & \longleftarrow & X_{i_k} & \supset & \partial X_{i_k}
\end{array}
\]

\[
\Sigma_{\{i_0, \dots, i_k\}}(\mathcal{X}) = \Sigma(\Delta_{i_0 \dots i_k}, \partial\Delta_{i_0 \dots i_k})
\]

\[
\begin{cases}
X_i \text{ est loc.\ fermé} \\
\operatorname{Int} X_i = X - \overline{U_i} = X_i \setminus C_i \\
\overline{X_i} = X_i \cup B'_i \qquad (B'_i = \dot{K} \setminus B_i)
\end{cases}
\]

\[
X_i = \bigl(\overline{W} \cap X_i = W \cup B_i\bigr) \amalg_{B_i} \overline{U_i}
\]

\[
\begin{cases}
U = U_K \setminus B = \operatorname{Int}(U_K) \\
V = X \setminus \overline{U_K} = (X \setminus U_K) \setminus C = \operatorname{Int}(X_1),
\qquad X_1 = X \setminus U_K \\
U = \complement\overline{V}, \quad V = \complement\overline{U}, \quad
\dot{U} = \dot{V} = \overline{U} \cap \overline{V} = \complement(U \cup V) = B \cup C \\
\struck{U \cap K = U \subset K,\ V \cap K = V_K}\quad V \subset X_1 \subset \overline{V}
\end{cases}
\]

\[
\begin{aligned}
t(\lambda_0) &= \lambda_0 \\
t(\lambda_1) &= \lambda_1 \lambda_2 \lambda_1^{-1} = \lambda_0^{-1} \lambda_2 \lambda_0 \\
t(\lambda_2) &= \lambda_1
\end{aligned}
\]

\[
\begin{aligned}
t^3(\lambda_0) &= \lambda_0 \\
t^3(\lambda_1) &= (\lambda_1 \lambda_2 \lambda_1^{-1})\, \lambda_1\, (\lambda_1 \lambda_2^{-1} \lambda_1^{-1})
= \lambda_1 \lambda_2 \lambda_1 \lambda_2^{-1} \lambda_1^{-1} \\
&= \operatorname{int}(\lambda_1 \lambda_2)\, \lambda_1 = \operatorname{int}(\lambda_0^{-1})\, \lambda_1 \\
t^3(\lambda_2) &= t(\lambda_1) = \lambda_1 \lambda_2 \lambda_1^{-1} = \struck{\operatorname{int}(\lambda_1)} \\
&= (\lambda_1 \lambda_2)\, \lambda_2\, (\lambda_2^{-1} \lambda_1^{-1})
= \operatorname{int}(\lambda_1 \lambda_2)\, \lambda_2\ \operatorname{int}\ill{}
\end{aligned}
\]

\begin{align*}
B' &= U'_K \cap \dot{K} \\
U' &= U'_K \setminus B' = \operatorname{Int}(U'_K) \\
V' &= X \setminus \overline{U'_K} = (X \setminus U'_K) \setminus C = \operatorname{Int}(X'_1),
\qquad X'_1 = X \setminus U'_K
\end{align*}

\[
\begin{cases}
\overline{U} \cap \overline{U'} = C, \quad \overline{U} \cap \overline{W} = B, \quad \overline{W} \cap \overline{U'} = B' \\
\dot{U}' = \overline{U'} \setminus U' = B' \cup C, \quad \dot{U} = B \cup C, \quad \dot{W} = B \cup B' \\
K = \overline{U \cup U'} = \complement W = (U \cup U') \cup B \cup B' \cup C, \quad
\operatorname{Int} K = U \cup U' \cup C
\end{cases}
\]

\[
\begin{cases}
\overline{U_K} = K \setminus U'_K, \quad \overline{\uncertain{U'_K}} = K \setminus U_K \\
C \cap \dot{K} = \emptyset \quad \text{car} \quad C = \overline{U_K} \cap \overline{U'_K} = K \setminus (U_K \cup U'_K)
\end{cases}
\]

\[
D(\{i\}) = \pi_0 \Sigma(X_i, \partial X_i) \xrightarrow[\sim]{\ \alpha_i\ } E(\{i\}) = \varphi^{-1}(\{i\})
\]

\[
\partial X_i = \bigcup_{\mu(\varphi(M(e))) < i} |E_e|
\]

\[
\partial X_i \overset{\mathrm{df}}{=} \bigcup_{j < i} X_j
= \bigcup_{j < i}\ \bigcup_{\substack{e \in E \\ \varphi M(e) \leq j}} |E_e|
= \bigcup_{\substack{e \in E \\ \exists j \text{ tel que } \varphi(M(e)) \leq j < i}} |E_e|
\]

\[
|K| = (K_0, K) \supset (L_0, L) = |L|, \qquad \Sigma(|K|, |L|) = |\Delta|
\]

\[
\begin{cases}
\tilde{K} \overset{\mathrm{df}}{=} \Sigma(K, L) = (\tilde{K}_0, \tilde{K}) \\
\tilde{K}_0 = \varinjlim_{\sigma \in K \setminus L} \sigma \ (\subset K_0) \xrightarrow{\ \pi\ } K_0 \\
\tilde{\Delta} = \operatorname{Im}\bigl((K \setminus L) \to \mathcal{P}(\Delta_0)\bigr)
\end{cases}
\]

\[
\Gamma_1 = \Gamma'_2 \cup \Gamma'_3, \qquad
\Gamma_2 = \Gamma'_3 \cup \Gamma'_1, \qquad
\Gamma_3 = \Gamma'_1 \cup \Gamma'_2
\]

\[
B \simeq I \times \{0, 1\}, \quad V_i :
\qquad
U_i = \mathring{\Gamma}_i \cup \bigcup_{j \in I \setminus \{i\}} U_j,
\qquad
V_i = \bigcap_{j \in I \setminus \{i\}} V_j
\]

\[
\left\{
\begin{array}{l}
V_1, V_2, V_3 \\
\Gamma^{\circ}_1, \Gamma^{\circ}_2, \Gamma^{\circ}_3 \\
\partial
\end{array}
\right.
\quad
\left\{
\begin{array}{l}
U_1 = V_2 \cup V_3 \cup \Gamma^{\circ}_1 \\
U_2 = V_3 \cup V_1 \cup \Gamma^{\circ}_2 \\
U_3 = V_1 \cup V_2 \cup \Gamma^{\circ}_3
\end{array}
\right.
\qquad
\begin{array}{l}
V_3 = U_1 \cap U_2 \\
V_2 = U_3 \cap U_1 \\
V_1 = U_1 \cap U_2
\end{array}
\quad
\left\{
\begin{array}{l}
\Gamma_1 = \Gamma^{\circ}_2 \cup \Gamma^{\circ}_3 \cup \partial \\
\Gamma_2 = \Gamma^{\circ}_3 \cup \Gamma^{\circ}_1 \cup \partial \\
\Gamma_3 = \Gamma^{\circ}_1 \cup \Gamma^{\circ}_2 \cup \partial
\end{array}
\right.
\]

\begin{align*}
\operatorname{Int}\Gamma'' &= \operatorname{Int}\Gamma \cup \operatorname{Int}\Gamma' \cup (I - \partial I) \\
\operatorname{Ext}\Gamma'' &= \operatorname{Ext}\Gamma \cap \operatorname{Ext}\Gamma'
\end{align*}

\begin{align*}
\operatorname{Int}\Gamma'' &= \operatorname{Int}\Gamma \setminus \operatorname{Int}\Gamma' \setminus (\Gamma' - I)
\quad \bigl[= \operatorname{Int}\Gamma \cap \operatorname{Ext}\Gamma'\bigr] \\
\operatorname{Ext}\Gamma'' &= \operatorname{Ext}\Gamma \cup \operatorname{Int}\Gamma' \cup (I \setminus \partial I)
\end{align*}

\[
\left\{
\begin{array}{l}
\Gamma^1_{xy} \cap \Gamma^2_{xy} = \{x,y\} \\[2pt]
\Gamma^1_{x,y} \cup \Gamma^2_{x,y} = \Gamma \\[2pt]
S'(\Gamma^1_{xy}) \cup S'(\Gamma^2_{xy}) = S(\Gamma) \smallsetminus \{x,y\}
\end{array}
\right.
\]