\[
  (1)\qquad \mathrm{Hom}_{\mathrm{isot}}(X,Y)=\text{ensemble des composantes connexes}
\]

\[
  (2)\qquad F\colon X\times I\longrightarrow Y\times I,\qquad I=[0,1]
\]

\[
  (3)\qquad F(x,0)=(f(x),0),\qquad F(x,1)=(g(x),1)\qquad\forall x\in X .
\]

\[
  Y'\longmapsto Y'\times I
\]

\[
  (4)\qquad \alpha\colon X\times I\xrightarrow{\ \sim\ }Y'\times I
\]

\[
  (6)\qquad f=pf_{0},\qquad g=pg_{0}\qquad (p\colon Y'\to Y\ \text{la projection}),
\]

\[
  (7)\qquad f_{0}\colon X\xrightarrow{\ \sim\ }Y',\qquad g_{0}\colon X\xrightarrow{\ \sim\ }Y'
\]

\[
  (8)\qquad g_{0}=f_{0}\circ u,
\]

\[
  (9)\qquad u\colon X\xrightarrow{\ \sim\ }X
\]

\[
  g=pg_{0}=p(f_{0}\circ u)=(pf_{0})u=f\circ u .
\]

\[
  (10)\qquad X\xrightarrow[\ \sim\ ]{\ f_{0}\ }Y'\xrightarrow{\ p\ }Y
\]

\[
  (11)\qquad (\gamma f_{0}\sim f_{0})\Longrightarrow\gamma=\mathrm{id}_{Y'}\ ?
\]

\[
  (12)\qquad \widehat{\mathrm{Hom}}_{\mathrm{ét}}(X,Y)=\mathrm{Hom}_{\mathrm{ét}}(X,Y)\wedge^{T_{X}}\widehat{T}_{X}
\]

\[
  (13)\qquad
  \begin{cases}
    \widehat{\mathrm{Hom}}_{\mathrm{ét}}(X,Y)\times\widehat{\mathrm{Hom}}_{\mathrm{ét}}(Y,Z)\longrightarrow\widehat{\mathrm{Hom}}_{\mathrm{ét}}(X,Z)\\
    \qquad (u,v)\longmapsto v\circ u
  \end{cases}
\]

\[
  (14)\qquad
  \begin{cases}
    \mathrm{Hom}(X,Y)\times\mathrm{Hom}(Y,Z)\longrightarrow\mathrm{Hom}(X,Z)\\
    \qquad (u,v)\longmapsto vu
  \end{cases}
\]

\[
  (g,f,g',f')\longmapsto (g'vf'^{-1})\circ(guf^{-1}),
\]

\[
  \mathrm{Aut}(Y)\times\mathrm{Aut}(X)\times\mathrm{Aut}(Z)\times\mathrm{Aut}(Y)\longrightarrow\mathrm{Hom}(X,Z)
\]

\[
  (15)\qquad \mathrm{Aut}(Y)\longrightarrow\mathrm{Hom}(X,Z),\qquad \varphi\longmapsto v\varphi u
\]

\[
  G_{u}\subset\mathrm{Aut}(X)\times\mathrm{Aut}(Y),\qquad
  G_{u}=\{\psi,\varphi \mid \varphi u\psi^{-1}=u \text{ i.e. } \varphi u=u\psi\},
\]

\[
  G_{u}\xrightarrow{\ \mathrm{pr}_{2}\ }\mathrm{Aut}(Y)\longrightarrow\mathrm{Hom}(X,Z)
\]

\[
  v(\alpha_{i}\varphi)u=v\alpha_{i}u\psi .
\]

\[
  (17)\qquad \mathrm{Hom}_{\widehat{C}_{0}}(X,Y)=\mathrm{Hom}_{C_{0}}(X,Y)\wedge^{\mathrm{Aut}(X)\times\mathrm{Aut}(Y)^{\circ}}\bigl(\widehat{\mathrm{Aut}}(X)\times\widehat{\mathrm{Aut}}(Y)\bigr)
\]

\[
  \simeq\ \mathrm{Hom}_{C_{0}}(X,Y)\wedge^{\mathrm{Aut}(X)}\widehat{\mathrm{Aut}}(X)
\]

\[
  (18)\qquad \hat{f}\colon X\longrightarrow Y
\]

\[
  (19)\qquad X\xrightarrow[\ \sim\ ]{\ \hat{f}_{0}\ }Y'\xrightarrow{\ u\ }Y
\]

\[
  \mathrm{Aut}_{Y}(Y')\longrightarrow\mathrm{Aut}(Y')\longrightarrow\mathrm{Aut}(Y')^{\wedge}
\]

\[
  (20)\qquad Y\xrightarrow[\ \sim\ ]{\ \hat{g}_{0}\ }Z'\xrightarrow{\ v\ }Z,
\]

\[
  \hat{g}_{0}=w\hat{\varphi},\qquad \hat{\varphi}\in\hat{H}_{u}\subset\widehat{\mathrm{Aut}}_{C_{0}}(Y)
\]

\[
  (22)\qquad \hat{g}\hat{f}=(v\hat{g}_{0})(u\hat{f}_{0})=(vw\hat{\varphi})(u\hat{f}_{0})=\underbrace{(vwu)}_{\in\,\mathrm{Fl}\,C_{0}}\underbrace{(\hat{\varphi}'\hat{f}_{0})}_{\in\,\mathrm{Fl\,iso.}(\widehat{C}_{0})}
\]

\[
  (23)\qquad X=Y\setminus T\hookrightarrow Y
\]

\[
  (24)\qquad
  \begin{array}{ccc}
    X & \hookrightarrow & \hat{X}\\
    \downarrow & & \downarrow\\
    Y & \hookrightarrow & \hat{Y}
  \end{array}
\]

\[
  (25)\qquad S\supset\hat{X}\,|\,T
\]

\[
(1)\quad
\begin{cases}
a_i^{\omega} : R_i^{\omega} \to R_i^{\omega'} \\
b_i^{\omega} : R^{\omega}_{\omega i} \to R^{\omega'}_{\omega' i}
\end{cases}
\]

\[
(2)\quad
\begin{cases}
\text{a)}\ \ b_i^{\omega} b_i^{\omega'} = 1 \\
\text{b)}\ \ a_i^{\omega} b^{\omega}_{\omega' i} a_i^{\omega'} b^{\omega}_{\omega i} a^{\omega'}_{\omega i} b_i^{\omega} = 1
\end{cases}
\]

\[
(3)\quad a_i^{\omega'} a_i^{\omega} = \lambda_i^{\omega} = \varphi_i^{\omega}(l_0)
\]

\[
(4)\quad \varphi_i^{\omega} : \Pi_{0,3} = \pi_1(M_{0,3}\,;R_0^{+}) \hookrightarrow \pi_1(\Sigma^{*}, R_i^{\omega})
\]

\[
\Pi_{0,3} = \{\, l_0, l_1, l_\infty \mid l_\infty l_1 l_0 = 1 \,\}
\subset \widetilde{\Pi}_{0,3} = \pi_1(M_{0,3}, \mathfrak{S}_3\,; R_0^{+}),
\]

\[
l_0 \mapsto a_i^{\omega'} a_i^{\omega}, \qquad
l_1 \mapsto (b^{\omega}_{\omega' i})^{-1} a^{\omega}_{\omega i} a^{\omega'}_{\omega i} b_i^{\omega}
\]

\[
(5)\quad \tau\varphi_i^{\omega}(g) = \varphi_i^{\omega}(\tau_\infty(g)) \quad \text{où } \tau_\infty \in \widetilde{\Pi}_{0,3},
\]

\[
\tau_\infty(l_0) = l_0^{-1}, \quad \tau_\infty(l_1) = l_1^{-1}, \quad
\tau_\infty(l_\infty) = l_0 l_1 = \mathrm{int}(l_0)\, l_\infty^{-1}
\]

\[
(6)\quad
\begin{cases}
a_i^{\omega}(\varphi_i^{\omega}(g)) = \varphi_i^{\omega'}(\varepsilon_0(g)) \\
b_i^{\omega}(\varphi^{\omega}_{\omega i}(g)) = \varphi^{\omega'}_{\omega' i}(\sigma_\infty(g))
\end{cases}
\qquad \varepsilon_0^2 = l_0
\]

\[
\sigma_\infty(l_0) = l_1, \quad \sigma_\infty(l_1) = l_0, \quad
\sigma_\infty(l_\infty) = (l_0 l_1)^{-1} = l_1^{-1} l_0^{-1} = \mathrm{int}(l_0)\, l_\infty
\]

\[
(7)\quad \tilde b_i^{\omega} = a^{\omega'}_{\omega' i} b_i^{\omega} : R^{\omega}_{\omega i} \to R^{\omega}_{\omega' i} ;
\]

\[
(8)\quad \tilde b^{\omega}_{\omega^2 i}\, \tilde b^{\omega}_{\omega i}\, \tilde b_i^{\omega} = 1
\]

\[
(9)\quad \tilde b_i^{\omega}(\varphi^{\omega}_{\omega i}(g)) = \varphi^{\omega}_{\omega' i}(\rho^{-1}(g))
\qquad \rho^{-1} = \varepsilon_0 \sigma_\infty,
\quad
\begin{cases}
\rho(l_0) = l_1 \\
\rho(l_1) = l_\infty \\
\rho(l_\infty) = l_0
\end{cases}
\]

\[
(\text{\struck{\ill{}}})\quad \tilde b_{\omega^2 i}\, \tilde b_{\omega i}\, \tilde b_i = 1
\]

\[
u(a_i^{\omega}) = a_i^{\omega} \varphi_i^{\omega}(l_0^{\gamma}) = \varphi_i^{\omega'}(l_0^{\gamma})\, a_i^{\omega}
\qquad \text{($\varphi_i^{\omega}(l_0^{\gamma}) = (\lambda_i^{\omega})^{\gamma}$)}
\]

\[
u(a_i^{\omega'} a_i^{\omega}) = \varphi_i^{\omega}(l_0^{2\gamma+1}), \qquad
u(\underbrace{\varphi_i^{\omega}(l_0)}_{\lambda_i^{\omega}}) = \varphi_i^{\omega}(l_0^{p}) = (\lambda_i^{\omega})^{p},
\qquad \gamma = \frac{p-1}{2}
\]

\[
\begin{cases}
u(a_i^{\omega}) = a_i^{\omega} \varphi_i^{\omega}(l_0^{\gamma}) \\
u(b_i^{\omega}) = b_i^{\omega} \varphi^{\omega}_{\omega i}(\beta)
\quad (= \varphi^{\omega'}_{\omega' i}(\sigma_\infty \beta)\, b_i^{\omega})
\end{cases}
\qquad \gamma = \frac{p-1}{2}, \quad p \in \hat{\mathbb{Z}}^{*} \text{ multiplicateur}, \quad \beta \in \widehat{\Pi}_{0,3}
\]

\[
u(\tilde b_i^{\omega}) = \tilde b_i^{\omega} \varphi^{\omega}_{\omega i}(\underbrace{l_1^{\gamma} \beta}_{\sigma_\infty(\lambda) \beta})
= \varphi^{\omega'}_{\omega' i}(l_0^{\gamma} \rho^{-1}(\beta))\, \tilde b_i^{\omega}
\]

\[
(12)\quad
\begin{cases}
\beta\, \sigma_\infty(\beta) = 1 \\
\alpha\, \rho^{-1}(\alpha)\, \rho^{-2}(\alpha) = 1
\quad \text{où } \text{\struck{$\alpha = l_1^{-\gamma} \beta$}} \\
\alpha = \underbrace{l_1^{\gamma}}_{\sigma_\infty(\lambda)} \beta
\end{cases}
\]

\[
Q_i \xrightarrow{\;c_i^{\omega}\;} R^{\omega'}_{\omega' i}
\]

\[
(13)\quad b_i^{\omega} = c_i^{\omega} (c_i^{\omega'})^{-1}
\]

\[
(14)\quad \tilde a_i^{\omega} = (c^{\omega}_{\omega i})^{-1} a_i^{\omega} c^{\omega'}_{\omega' i} : Q_{\omega i} \to Q_{\omega' i}
\]

\[
(15)\quad \tau(\tilde a_i^{\omega}) = (\tilde a_i^{\omega'})^{-1}
\]

\[
(16)\quad \tilde a^{\omega}_{\omega^2 i}\, \tilde a^{\omega}_{\omega i}\, \tilde a_i^{\omega} = 1
\]

\[
(17)\quad \psi_i^{\omega} : \Pi_{0,3} \xrightarrow{\ \sim\ } \pi(Q_i)
\]

\[
(18)\quad \psi_i^{\omega}(g) = \psi_i^{\omega'}(\sigma_\infty(g))
\]

\[
(19)\quad
\begin{aligned}
\tilde a_i^{\omega}(\psi^{\omega'}_{\omega i}(g)) &= \psi^{\omega}_{\omega' i}(\varepsilon_0(g)) = \psi^{\omega'}_{\omega' i}(\sigma_\infty \varepsilon_0(g)) \\
\tilde a_i^{\omega}(\psi^{\omega}_{\omega i}(g)) &= \psi^{\omega}_{\omega' i}(\underbrace{(\varepsilon_0 \sigma_\infty)}_{\rho^{-1}}(g)) = \psi^{\omega'}_{\omega' i}(\sigma_\infty \varepsilon_0 \sigma_\infty(g))
\end{aligned}
\]

\[
(20)\quad u(a_i^{\omega}) = a_i^{\omega} \varphi_i^{\omega}(l_0^{\gamma})
\qquad \gamma = \frac{p-1}{2} \in \hat{\mathbb{Z}}, \quad p \in \hat{\mathbb{Z}}^{*}
\]

\[
(21)\quad u(c_i^{\omega}) = \varphi^{\omega'}_{\omega' i}(\eta)\, c_i^{\omega} = c_i^{\omega} \psi_i^{\omega}(\eta)
\qquad \eta \in \widehat{\Pi}_{0,3} \text{ indépendant de } i, \omega
\]

\[
(22)\quad u(b_i^{\omega}) = b_i^{\omega} \varphi^{\omega}_{\omega i}(\beta)
\]

\[
(23)\quad \beta = \sigma_\infty(\eta)\, \eta^{-1}
\]

\[
(24)\quad \text{\struck{$l_1^{-\gamma}\sigma_\infty(\eta)\eta^{-1}$}}\ \ \alpha \rho^{-1}(\alpha) \rho^{-2}(\alpha) = 1
\quad \text{avec } \alpha = \underbrace{l_1^{\gamma}}_{\sigma_\infty(\lambda)} \underbrace{\sigma_\infty(\eta)\eta^{-1}}_{\beta}.
\]

\[
(25)\quad u(\tilde a_i^{\omega}) = \psi^{\omega^*}_{\omega i}(\underbrace{\eta^{-1} l_0^{\gamma} \varepsilon_0(\eta)}_{\tilde\alpha = \eta^{-1}\varepsilon_0(\lambda\eta)})\, \tilde a_i^{\omega}
= \tilde a_i^{\omega}\, \psi^{\omega'}_{\omega' i}(\underbrace{\varepsilon_0^{-1}(\eta^{-1}) l_0^{\gamma} \eta}_{\varepsilon_0^{-1}(\tilde\alpha)})
\]

\[
(\text{25\,bis})\quad \alpha = \rho(\eta \tilde\alpha \rho^{-1}(\eta)^{-1})
\]

\[
(26)\quad \tilde\alpha\, \rho^{-1}(\tilde\alpha)\, \rho^{-2}(\tilde\alpha) = 1, \qquad
\boxed{(27)\quad \tilde\alpha = \eta^{-1} l_0^{\gamma} \varepsilon_0(\eta)}
\]

\[
(27)\quad \tau Q_i = Q_i, \qquad \tau \tilde a_i^{\omega} = (\tilde a_i^{\omega'})^{-1}
\]

\[
(28)\quad \tilde\lambda_i^{\omega} = \tilde a_i^{\omega'} \tilde a_i^{\omega} \in \pi(Q_{\omega' i})
\qquad (\lambda_i^{\omega} = a_i^{\omega'} a_i^{\omega})
\]

\[
(29)\quad u(\tilde\lambda_i^{\omega}) = \psi^{\omega'}_{\omega' i}(\tilde\alpha\, \varepsilon_0(\tilde\alpha))\, \tilde\lambda_i^{\omega}
= \psi^{\omega'}_{\omega' i}(\tilde\alpha\, \varepsilon_0(\tilde\alpha)\, l_0)
\]

\[
(30)\quad \tilde\alpha\, \varepsilon_0(\tilde\alpha)\, l_0 = \eta^{-1} l_0^{p} \eta \qquad (p = 2\gamma+1) ;
\]

\[
(31)\quad \tilde\alpha' = \varepsilon_0 \tau_\infty(\tilde\alpha^{-1}) = \underbrace{\rho^{-1} \tilde\sigma_\infty}_{\tilde\sigma_0}(\tilde\alpha^{-1})
\]

\[
\tilde\sigma_i(l_i) = l_i^{-1}, \qquad
\tilde\sigma_i(l_j) = l_k^{-1} \quad \text{si } \{0, 1, \infty\} = \{i, j, k\}
\]

\[
x\, \rho^{-1}(x)\, \rho^{-2}(x) = 1
\]

\[
(32)\quad \Theta_g x = g \cdot x^{\chi(g)}
\]

\[
(33)\quad \tilde\alpha' = \Theta_{\tilde\sigma_0}(\tilde\alpha).
\]

\[
d_i^{\omega} : P^{\omega} \to Q_i
\]

\[
(34)\quad \tau(P^{\omega}) = P^{\omega'}, \qquad \tau(d_i^{\omega}) = d_i^{\omega'}
\]

\[
(35)\quad (d^{\omega}_{\omega' i})^{-1} \tilde a_i^{\omega} d_i^{\omega} = 1
\]

\[
(36)\quad
\begin{cases}
u(\tilde a_i^{\omega}) = \psi^{\omega}_{\omega i}(\tilde\alpha)\, \tilde a_i^{\omega} & \tilde\alpha \in \widehat{\Pi}_{0,3} \ \text{(indép. de } i, \omega) \\
u(d_i^{\omega}) = \psi_i^{\omega}(\zeta)\, d_i^{\omega} \quad (= d_i^{\omega} \tilde\varphi_i^{\omega}(\zeta)) & \zeta \in \widehat{\Pi}_{0,3}\ \ldots
\end{cases}
\]

\[
(37)\quad \tilde\alpha = \zeta \cdot \rho^{-1}(\zeta^{-1})
\]

\[
(39)\quad \tilde\psi^{\omega}_{\omega i}(g) = \tilde\psi_i^{\omega}(\rho g) \quad \text{i.e.} \quad \tilde\psi_{\omega i} = \tilde\psi_i \circ \rho,
\]

\[
(40)\quad \tilde d_i^{\omega} : P^{\omega} \to P^{\omega'}, \qquad \tilde d_i^{\omega} = (d_i^{\omega'})^{-1} d_i^{\omega},
\]

\[
(41)\quad \tau(\tilde d_i^{\omega}) = \tilde d_i^{\omega'}, \qquad \tilde d_i^{\omega} \tilde d_i^{\omega'} = 1
\]

\[
(42)\quad \tilde d_i^{\omega}(\tilde\psi_i^{\omega}(g)) = \tilde\psi_i^{\omega'}(\sigma_\infty g).
\]

\[
(43)\quad u(\tilde d_i^{\omega}) = \tilde\psi_i^{\omega'}(\sigma_\infty \tilde\beta)\, \tilde d_i^{\omega} = \tilde d_i^{\omega}\, \tilde\psi_i^{\omega}(\tilde\beta)
\]

\[
(44)\quad \tilde\beta = \zeta\, \sigma_\infty(\zeta)^{-1}.
\]

\[
(45)\quad \tilde\beta\, \sigma_\infty(\tilde\beta) = 1
\]

\[
(46)\quad u(\tilde d_i^{\omega}) = \tilde d_i^{\omega} \tilde\psi_i^{\omega}(\tilde\beta)
\qquad \text{avec } \tilde\beta \in \widehat{\Pi}_{0,3} \text{ indépendant de } i, \omega
\]

\[
(47)\quad \tilde\beta\, \sigma_\infty(\tilde\beta) = 1
\]

\[
(48)\quad (d^{\omega}_{\omega i})^{-1} (c^{\omega'}_{\omega i})^{-1} (a^{\omega}_{\omega i})^{-1} c_i^{\omega} d_i^{\omega} = 1
\]

\[
(49)\quad
\begin{cases}
\tau(R_i^{\omega}) = R_i^{\omega'}, \quad \tau Q_i = Q_i, \quad \tau(P^{\omega}) = P^{\omega'} \\
\tau(a_i^{\omega}) = (a_i^{\omega'})^{-1}, \quad \tau(c_i^{\omega}) = c_i^{\omega'}, \quad \tau(d_i^{\omega}) = d_i^{\omega'}
\end{cases}
\]

\[
(50)\quad
\begin{cases}
\Pi_{0,3} \xrightarrow{\ \varphi_i^{\omega}\ } \pi(R_i^{\omega}) \quad \text{\struck{$\to \pi($}} \\
\Pi_{0,3} \xrightarrow{\ \psi_i^{\omega}\ } \pi(Q_i) \\
\Pi_{0,3} \xrightarrow{\ \tilde\psi_i^{\omega}\ } \pi(P^{\omega})
\end{cases}
\]

\[
(51)\quad
\begin{cases}
d_i^{\omega}(\tilde\psi_i^{\omega}(g)) = \psi_i^{\omega}(g) \\
c_i^{\omega} \psi_i^{\omega}(g) = \varphi_i^{\omega}(g)
\end{cases}
\]

\[
(52)\quad \psi_i^{\omega}(g) = \psi_i^{\omega'}(\sigma_\infty(g))
\]

\[
(53)\quad a_i^{\omega} \varphi_i^{\omega}(g) = \varphi_i^{\omega'}(\varepsilon_0(g))
\]

\[
(54)\quad
\begin{cases}
u(a_i^{\omega}) = a_i^{\omega} \varphi_i^{\omega}(\lambda) \\
u(c_i^{\omega}) = \varphi^{\omega'}_{\omega' i}(\eta)\, c_i^{\omega} = c_i^{\omega} \psi_i^{\omega}(\eta) \\
u(d_i^{\omega}) = \psi_i^{\omega}(\zeta)\, d_i^{\omega} = d_i^{\omega} \tilde\psi_i^{\omega}(\zeta)
\end{cases}
\]

\[
(55)\quad \tilde\alpha = \eta^{-1} \varepsilon_0(\lambda\eta) \quad \text{et} \quad \tilde\alpha = \zeta \rho^{-1}(\zeta^{-1})
\]

\[
(56)\quad \eta^{-1} \varepsilon_0(\lambda\eta) = \zeta \rho^{-1}(\zeta)^{-1}
\]

\[
(57)\quad
\begin{cases}
\beta = \sigma_\infty(\eta)\eta^{-1} & (u(b_i^{\omega}) = b_i^{\omega} \varphi^{\omega}_{\omega i}(\beta)) \\
\alpha = \sigma_\infty(\lambda)\beta & (u(\tilde b_i^{\omega}) = \tilde b_i^{\omega} \varphi^{\omega}_{\omega i}(\alpha)) \\
\tilde\alpha = \eta^{-1}\varepsilon_0(\lambda\eta) = \zeta\rho^{-1}(\zeta)^{-1} & (u(\tilde a_i^{\omega}) = \psi^{\omega}_{\omega i}(\tilde\alpha)\, \tilde a_i^{\omega})
\end{cases}
\]

\[
(58)\quad
\begin{cases}
\beta' = \beta^{-1} \ (= \sigma_\infty(\beta)) = \eta\, \sigma_\infty(\eta)^{-1} \\
\alpha' = \alpha^{-1} = \beta^{-1} \sigma_\infty(\lambda)^{-1} = \beta' \sigma_\infty(\lambda^{-1})
\end{cases}
\qquad \text{donc } \beta' = \alpha' \sigma_\infty(\lambda)
\]

\[
(59)\quad
\begin{aligned}
\beta' &= \eta\, \sigma_\infty(\eta)^{-1} \quad \text{\struck{\ill{}}} \\
\alpha' &= \text{\struck{\ill{}}}\ \beta' \sigma_\infty(\lambda^{-1}) \quad \text{i.e.} \quad \beta' = \alpha' \sigma_\infty(\lambda)
\end{aligned}
\]

\[
(60)\quad
\begin{cases}
\alpha' = \zeta' \rho^{-1}(\zeta')^{-1} \\
\text{i.e.}\quad \eta\, \sigma_\infty(\eta)^{-1} = (\zeta' \rho^{-1}(\zeta')^{-1})\, \sigma_\infty(\lambda)
\end{cases}
\]

\[
\alpha = \rho(\eta \tilde\alpha \rho^{-1}(\eta)^{-1})
\]

\[
\tilde\alpha' = \tilde\alpha^{-1}
\]

\[
(61)\quad \alpha' = \rho(\rho^{-1}(\eta)\, \tilde\alpha'\, \eta^{-1}) = \eta\, \rho(\tilde\alpha')\, \rho(\eta)^{-1}
\]

\[
\tilde\alpha' = \rho^{-1}(\zeta)\, \zeta^{-1}
\]

\[
\rho(\tilde\alpha') = \zeta\, \rho(\zeta)^{-1}
\]

\[
\alpha' = \eta\zeta\, \rho(\zeta^{-1}\eta^{-1}) \qquad \text{\struck{i.e.\ $\alpha'$ \ill{} (60)}}
\]

\[
\zeta' = \eta\zeta.
\]

\[
(62)\quad \{(\lambda, \eta, \zeta') \in \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3}\}
\]

\[
(63)\quad
\boxed{
\begin{aligned}
&\beta' = \alpha' \sigma_\infty(\lambda) \qquad (= \alpha' l_1^{\gamma} \ \text{si}\ \lambda = l_0^{\gamma}) \\
&\text{avec}\quad \alpha' = \zeta' \rho^{-1}(\zeta')^{-1} \\
&\phantom{\text{avec}\quad} \beta' = \eta\, \sigma_\infty(\eta)^{-1}
\end{aligned}
}
\]

\[
(64)\quad b_i'^{\omega} = (b_i^{\omega})^{-1} \ (= b_i^{\omega'}), \qquad \tilde b_i'^{\omega} = (\tilde b_i^{\omega})^{-1}, \qquad \tilde a_i'^{\omega} = (\tilde a_i^{\omega})^{-1}
\]

\[
(65)\quad
\begin{cases}
u(b_i'^{\omega}) = \varphi^{\omega}_{\omega i}(\beta')\, b_i'^{\omega} \\
u(\tilde b_i'^{\omega}) = \varphi^{\omega}_{\omega i}(\alpha')\, \tilde b_i'^{\omega} \\
u(\tilde a_i'^{\omega}) = \tilde a_i'^{\omega}\, \psi^{\omega}_{\omega i}(\tilde\alpha')
\end{cases}
\]

\[
(66)\quad \tilde d_i'^{\omega} : P^{\omega} \to R_i^{\omega'}, \qquad \tilde d_i'^{\omega} = c^{\omega}_{\omega i} \cdot d^{\omega}_{\omega i}
\]

\[
(67)\quad u(\tilde d_i'^{\omega}) = \varphi_i^{\omega'}(\underbrace{\zeta'}_{\eta\zeta})\, \tilde d_i'^{\omega}
\]

\[
(66')\quad d_i^{\omega} : P^{\omega} \to R_i^{\omega}, \qquad d_i^{\omega} = c^{\omega'}_{\omega' i} \cdot d^{\omega}_{\omega' i}
\]

\[
(67')\quad u(\tilde d_i^{\omega}) = \varphi_i^{\omega}(\eta\, \sigma_\infty(\zeta))\, \tilde d_i^{\omega}
\]

\[
(69)\quad (\lambda, \beta', \zeta') \in \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3}
\]

\[
(70)\quad
\begin{cases}
\beta' = \alpha' \sigma_\infty(\lambda) \quad \text{où} \quad \alpha' = \zeta' \rho^{-1}(\zeta')^{-1} \\
\beta' \sigma_\infty(\beta') = 1 \quad \text{i.e.} \quad \alpha' \sigma_\infty(\lambda)\, \sigma_\infty(\alpha')\, \lambda = 1
\end{cases}
\]

\[
(69')\quad (\lambda, \eta, \alpha') \in \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3} \times \widehat{\Pi}_{0,3}
\]

\[
(70')\quad
\begin{cases}
\beta' = \alpha' \sigma_\infty(\lambda) \quad \text{où} \quad \beta' = \eta\, \sigma_\infty(\eta)^{-1} \\
\alpha' \rho(\alpha') \rho^{2}(\alpha') = 1
\end{cases}
\]

\[
(71)\quad
\begin{cases}
\tilde d_i^{\omega} : P^{\omega} \to R_i^{\omega} \\
\tilde d_i'^{\omega} : P^{\omega} \to R_i^{\omega'} \\
a_i^{\omega} : R_i^{\omega} \to R_i^{\omega'}
\end{cases}
\]

\[
(72)\quad
\begin{cases}
(\tilde d_i'^{\omega})^{-1} a_i^{\omega} \tilde d_i^{\omega} = 1 \\
(\tilde d'^{\omega}_{\omega i})^{-1} \tilde d^{\omega'}_{\omega i} (\tilde d_i'^{\omega'})^{-1} \tilde d_i^{\omega} = 1
\end{cases}
\]

\[
(73)\quad
\begin{cases}
u(\tilde d_i^{\omega}) = \varphi_i^{\omega}(\zeta_1)\, \tilde d_i^{\omega} \\
u(\tilde d_i'^{\omega}) = \varphi_i^{\omega'}(\zeta')\, \tilde d_i'^{\omega} \\
u(a_i^{\omega}) = a_i^{\omega} \varphi_i^{\omega}(\lambda)
\end{cases}
\qquad \zeta', \zeta_1, \lambda \in \widehat{\Pi}_{0,3}
\]

\[
(74)\quad \zeta' = \varepsilon_0(\lambda\zeta_1) \qquad \text{i.e.} \qquad \zeta_1 = \lambda^{-1} \cdot \varepsilon_0^{-1}(\zeta')
\]

\[
(75)\quad \beta\, \zeta_1 = \beta' \sigma_\infty(\zeta')
\]

\[
(75)\quad
\begin{aligned}
\beta' &= \zeta_1 \sigma_\infty(\zeta')^{-1} = \text{\struck{$\varepsilon_0(\lambda\zeta_1)$}}\ \zeta_1 (\sigma_\infty\varepsilon_0)(\lambda\zeta_1)^{-1} \\
&= \lambda^{-1} \cdot \varepsilon_0^{-1}(\zeta')\, \sigma_\infty(\zeta')^{-1}.
\end{aligned}
\]

\[
(75')\quad \sigma_\infty(\beta') = \sigma_\infty(\zeta_1)\, \zeta'^{-1} = \sigma_\infty(\zeta_1)\, \varepsilon_0(\lambda\zeta_1)^{-1}
= \sigma_\infty(\lambda)^{-1} \overbrace{\rho(\zeta')\, \zeta'^{-1}}^{\alpha'^{-1}}
\]

\[
\text{\struck{$\beta' \sigma_\infty(\beta') = 1$}}
\]

\[
(77)\quad \sigma_\infty(\beta')^{-1} = \alpha' \sigma_\infty(\lambda).
\]

\[
(78)\quad \beta' \sigma_\infty(\beta') = 1 \qquad \text{ou encore} \quad \sigma_\infty(\beta')^{-1} = \beta'
\]

\[
\beta_1 = \sigma_\infty(\beta')^{-1} = \alpha' \sigma_\infty(\lambda)
\]

\[
(\ast\ast)\quad \beta_1 \sigma_\infty(\beta_1) = 1
\]

\[
(78)\quad \beta' = \alpha' \sigma_\infty(\lambda) \qquad \text{où} \quad \alpha' = \zeta' \rho(\zeta')^{-1}
\]

\[
  J \subset \Sigma, \qquad J = (R_i)_{i \in J}
\]

\[
  \Gamma_{\Sigma,J} = \Gamma \simeq \mathfrak{S}_J \times \{1, \tau\}
\]

\[
  (3)\quad
  \left\{
  \begin{array}{ll}
    \boldsymbol{\varpi} = \Omega(\Sigma) \simeq \Omega(E_3)
      & \text{ens. des deux orientations} \\
      & \text{de } \Sigma \text{, ou de } E \\
    \boldsymbol{\omega} = \Omega(J) \simeq \Omega(\Sigma_{\mathbf{R}}) & \\
    \boldsymbol{\varepsilon} = \pi_0(\Sigma \setminus \Sigma_{\mathbf{R}})
      = \pi_0(E \setminus V) = \pi_0\big((E/V)^{*}\big) &
  \end{array}
  \right.
\]

\[
  \Sigma_{\mathbf{R}} = \Sigma \cap V
  \qquad \text{cercle euclidien du plan euclidien } V .
\]

\[
  (4)\quad \sum_{j \in J} R_j = 0
\]

\[
  (5)\quad V \simeq V_J(\mathbf{R})
  = \operatorname{Ker}\big(\mathbf{R}^J \xrightarrow{\ \text{somme}\ } \mathbf{R}\big)
\]

\[
  (6)\quad \Delta = V^{\perp} \simeq \mathbf{R}(\boldsymbol{\varepsilon})
\]

\[
  (7)\quad E \simeq V_J(\mathbf{R}) \oplus \mathbf{R}(\boldsymbol{\varepsilon})
  \qquad \text{isom. d'espaces euclidiens}
\]

\[
  q(R_j) = 1
\]

\[
  (8)\quad g_E = \underbrace{g_{V_J(\mathbf{R})}}_{\text{tautol.}}
  \oplus \operatorname{sg}(g)\, \mathrm{id}_{\Delta}
\]

\[
  (9)\quad \tau_E = \mathrm{id}_{V_J(\mathbf{R})} \oplus (-\mathrm{id}_{\Delta})
\]

\[
  (10)\quad \Sigma^{*} = \Sigma \setminus J = \Sigma - \{R_i \mid i \in I\}
\]

\[
  (11)\quad \Sigma'_{R_i} \simeq P(T_{\Sigma,R_i})
\]

\[
  \chi_!(\Sigma) - 3\chi_!(\text{rondelle ouverte})
  + 3\big(\underbrace{\chi_!(\text{ruban Möbius})}_{0}\big)
  = 2 - 3 = -1 = 1 - g ,
\]

\[
  (12)\quad \partial\widetilde{\Sigma} \simeq \coprod_{i \in I} TU_{\Sigma,R_i}
\]

\[
  (13)\quad
  \operatorname{Rev\,et}(\widetilde{\Sigma})
  \xrightarrow[\text{équiv}]{\ \approx\ }
  \operatorname{Rev\,et}(\widetilde{\Sigma}^{\circ})
  \xrightarrow[\text{iso}]{\ \sim\ }
  \operatorname{Rev\,et}(\Sigma^{*})
\]

\[
  \widetilde{\Sigma}^{\circ}
  \overset{\mathrm{def}}{=} \widetilde{\Sigma} \setminus \partial\widetilde{\Sigma}
  = \widetilde{\Sigma} \,|\, \Sigma^{*} \hookrightarrow \widetilde{\Sigma}
\]

\[
  \widetilde{\Sigma}^{\circ} \xrightarrow{\ \sim\ } \Sigma^{*},
\]

\[
  (14)\quad \Pi_1(\widetilde{\Sigma}) \longrightarrow \Pi_1(\Sigma^{*})
\]

\[
  \Pi_1(U_i) \longrightarrow \Pi_1(\widetilde{\Sigma})
\]

\[
  (15)\quad \Pi_1\Big(\coprod U_i\Big) \longrightarrow \Pi_1(\Sigma^{*}) .
\]

\[
  \widetilde{\Sigma} \longrightarrow \Sigma, \qquad J \subset \Sigma
\]

\[
  (16)\quad
  \left\{
  \begin{array}{ll}
    Q_i \ (i \in J) & \text{antipodique de } R_i \ (= -R_i) \\
    R_i^{\omega} \in U_i & (i \in J,\ \omega \in \boldsymbol{\omega}) \\
    P_{\varepsilon} & (\varepsilon \in \boldsymbol{\varepsilon}
      = \pi_0(\Sigma \setminus \Sigma_{\mathbf{R}})) \\
    S_i^{\varepsilon} &
  \end{array}
  \right.
\]

\[
  (17)\quad \{P_\varepsilon\} = \Delta \cap (\Sigma - \Sigma_{\mathbf{R}})_\varepsilon
\]

\[
  (18)\quad
  U_{i\mathbf{R}} \overset{\mathrm{def}}{=}
  \underbrace{U_i \cap T_{\Sigma_{\mathbf{R}},R_i}}_{TU_{\Sigma_{\mathbf{R}},R_i}}
  \simeq \boldsymbol{\omega}
\]

\[
  (19)\quad S_i^{\varepsilon} \in U_i \cap T_{\Sigma_i} = TU_{\Sigma_i}
\]

\[
  J = \{0, 1, \infty\}, \qquad \boldsymbol{\omega} = \{+1, -1\}
\]

\[
  \boldsymbol{\varpi} \simeq \{+1, -1\}, \quad \text{d'où} \quad
  \boldsymbol{\varepsilon} \simeq \{+1, -1\} .
\]

\[
  \text{\struck{(19)}}\qquad
  \boldsymbol{\varpi} \simeq \{\pm 1\}
\]

\[
  (20)\quad
  \overset{12}{a_i^{\omega,\varepsilon}},\quad
  \overset{6}{b_i^{\omega}},\quad
  \overset{6}{c_i^{\varepsilon}},\quad
  \overset{6}{d_i^{\varepsilon}},
  \qquad i \in J,\ \omega \in \boldsymbol{\omega},\ \varepsilon \in \boldsymbol{\varepsilon}
\]

\[
  (22)\quad
  \tau a_i^{\omega,\varepsilon} = a_i^{\omega,\varepsilon'}, \quad
  \tau b_i^{\omega} = b_i^{\omega}, \quad
  \tau c_i^{\varepsilon} = c_i^{\varepsilon'}, \quad
  \tau d_i^{\varepsilon} = d_i^{\varepsilon'} .
\]

\[
  (23)\quad a_i^{\omega\varepsilon} : S_i^{\varepsilon} \longrightarrow R_i^{\omega}
  \quad \text{sur } U_i
\]

\[
  (24)\quad b_i^{\omega} : Q_i \longrightarrow R_{\omega' i}^{\omega'}
  \quad \text{sur } \widetilde{\Sigma}_{\mathbf{R}} \simeq \operatorname{Dic}(\Sigma_{\mathbf{R}}, J)
\]

\[
  (25)\quad c_i^{\omega} : P^{\omega} \longrightarrow Q_i
  \qquad \text{suivant un arc de grand cercle}
\]

\[
  (26)\quad d_i^{\omega} : P^{\omega} \longrightarrow S_i^{\omega}
  \qquad \text{suivant un arc de grand cercle}
\]

\[
  (27)\quad b_i : R_{\omega i}^{\omega'} \longrightarrow R_{\omega' i}^{\omega'}, \qquad
  \widetilde{b}_i^{\omega} = b_i^{\omega} (b_i^{\omega'})^{-1}
\]

\[
  (27)\quad
  \left\{
  \begin{array}{ll}
    \widetilde{\Sigma}_\varepsilon = \widetilde{\Sigma} \setminus
      \bigcup_{i \in J} \operatorname{supp} d_i^{\varepsilon}
      & \varepsilon \in \boldsymbol{\varepsilon} \\[4pt]
    \widetilde{\Sigma}_i = \widetilde{\Sigma} \setminus
      \bigcup_{j \in J \setminus \{i\}} b_j &
  \end{array}
  \right.
\]

\[
  (28)\quad
  \left\{
  \begin{array}{l}
    \Pi_\varepsilon = \pi_1(\widetilde{\Sigma}, \widetilde{\Sigma}_\varepsilon)
      \simeq \pi_1(\Sigma^{*}, \Sigma^{*}_\varepsilon)
      \simeq \pi_1(\Sigma^{*}, P_\varepsilon) \\[4pt]
    \Pi_i = \pi_1(\widetilde{\Sigma}, \widetilde{\Sigma}_i)
      = \pi_1(\Sigma^{*}, \Sigma^{*}_i)
      \simeq \pi_1(\Sigma^{*}, Q_i)
  \end{array}
  \right.
\]

\[
  (29)\quad
  \left\{
  \begin{array}{ll}
    \varphi_{\varepsilon,x} \text{ ou } \varphi_\omega :
      \Pi_\varepsilon \xrightarrow{\ \sim\ } \pi_1(\widetilde{\Sigma}, x)
      & \big(\simeq \pi_1(\Sigma^{*}, x) \text{ si } x \in \Sigma^{*}\big) \\[4pt]
    \varphi_{i,x} \text{ ou } \varphi_i :
      \Pi_i \xrightarrow{\ \sim\ } \pi_1(\widetilde{\Sigma}, x)
      & \text{—}
  \end{array}
  \right.
\]

\[
  (30)\quad
  \left\{
  \begin{array}{ll}
    \varphi_{i\varepsilon} : \Pi_\omega \longrightarrow \Pi_i
      & \text{défini par } c_i^{\varepsilon} : P_\varepsilon \to Q_i \\[4pt]
    \varphi_{\varepsilon i} : \Pi_i \longrightarrow \Pi_\omega
      & \text{---\ } (c_i^{\varepsilon})^{-1} : Q_i \to P_\varepsilon
  \end{array}
  \right.
\]

\[
  (31)\quad
  \left\{
  \begin{array}{ll}
    \varphi_{R_i^{\omega},\, \omega i} : \Pi_{\omega i} \xrightarrow{\ \sim\ }
      \pi_1(\widetilde{\Sigma}, R_i^{\omega})
      & \text{ou } \varphi_{\omega i} \\[4pt]
    \varphi_{S_i^{\varepsilon},\, \varepsilon} : \Pi_\varepsilon \xrightarrow{\ \sim\ }
      \pi_1(\widetilde{\Sigma}, S_i^{\varepsilon})
      & \text{ou } \varphi^{\varepsilon} \text{ ou } d_i^{\varepsilon}
  \end{array}
  \right.
\]

\[
  \text{\struck{$a_i^{\omega} = a_i^{\omega-} a_i^{\omega+}$, \quad $a_i^{\omega+} : R_i^{\omega} \to S_i^{\omega}$, \quad $a_i^{\omega-} : S_i^{\omega} \to R_i^{\omega'}$}}
\]

\[
  (32)\quad
  \Pi_{0,3} = \pi_1\big(\mathbf{P}^1_{\mathbf{C}} \setminus \{0,1,\infty\}
  \,[= U_{0,3}(\mathbf{C})],\ P^{+}\big)
  = \{\, l_0, l_1, l_\infty \mid l_\infty l_1 l_0 = 1 \,\}
\]

\[
  \subset \widetilde{\Pi}_{0,3}
  = \pi_1\big(U_{0,3}(\mathbf{C}),\ \mathfrak{S}_3 \times \{1,\tau\},\ P^{+}\big)
\]

\[
  = \{\, \rho, \sigma_\infty, \widetilde{\sigma}_\infty \mid
  \rho^3 = \sigma_\infty^2 = \widetilde{\sigma}_\infty^2 = 1,\
  \sigma_\infty(\rho) = \rho^{-1},\
  \widetilde{\sigma}_\infty(\sigma_\infty) = 1 \,\}
\]

\[
  \text{[i.e. } (\sigma_\infty \widetilde{\sigma}_\infty)^2 = 1 \text{]}
  \qquad \simeq \mathrm{GL}(2,\mathbf{Z})/\{\pm 1\}
\]

\[
  (33)\quad
  \left\{
  \begin{array}{l}
    \varepsilon_0 \overset{\mathrm{def}}{=} \sigma_\infty \rho, \quad
    \varepsilon_1 \overset{\mathrm{def}}{=} \rho(\varepsilon_0)
      \ (= \rho\, \varepsilon_0 \rho^{-1}), \\[4pt]
    \varepsilon_\infty \overset{\mathrm{def}}{=} \rho(\varepsilon_1)
      = \rho^2(\varepsilon_0) = \rho^{-1}(\varepsilon_0) \\[4pt]
    l_0 \overset{\mathrm{def}}{=} \varepsilon_0^2, \quad
    l_1 \overset{\mathrm{def}}{=} \varepsilon_1^2, \quad
    l_\infty \overset{\mathrm{def}}{=} \varepsilon_\infty^2, \quad
    l_\infty l_1 l_0 = 1 \\[4pt]
    \rho(l_i) = l_{\rho(i)}
      \ \ \{\rho(l_0) = l_1,\ \rho(l_1) = l_\infty,\ \rho(l_\infty) = l_0\}, \quad
    \rho(\varepsilon_i) = \varepsilon_{\rho(i)}
  \end{array}
  \right.
\]

\[
  (33)\quad
  \left\{
  \begin{array}{l}
    \tau_\infty \overset{\mathrm{def}}{=} \sigma_\infty \widetilde{\sigma}_\infty
      = \widetilde{\sigma}_\infty \sigma_\infty, \quad \tau_\infty^2 = 1 \\[4pt]
    \tau_\infty(l_0) = l_0^{-1}, \quad \tau_\infty(l_1) = l_1^{-1}, \quad
    \tau_\infty(l_\infty) = l_0^{-1} l_\infty^{-1} l_0 \\[4pt]
    \widetilde{\sigma}_\infty(l_0) = l_1^{-1}, \quad
    \widetilde{\sigma}_\infty(l_1) = l_0^{-1}, \quad
    \widetilde{\sigma}_\infty(l_\infty) = l_\infty^{-1} \\[4pt]
    \sigma_\infty(l_0) = l_1, \quad \sigma_\infty(l_1) = l_0, \quad
    \sigma_\infty(l_\infty) = l_1^{-1} l_0^{-1}
  \end{array}
  \right.
\]

\[
  (34)\quad
  \left\{
  \begin{array}{l}
    \psi_\varepsilon^{i,\omega} : \Pi_{0,3} \xrightarrow{\ \sim\ } \Pi_\varepsilon \\[4pt]
    \psi_i^{\omega,\varepsilon} : \Pi_{0,3} \xrightarrow{\ \sim\ } \Pi_i
  \end{array}
  \right.
\]

\[
  \text{\struck{(35)}}\quad
  \psi_r = \psi_{\varepsilon}^{i,\omega} : \Pi_{0,3} \xrightarrow{\ \sim\ } \Pi_\varepsilon
  \qquad r = (i, \omega, \varpi)
\]

\[
  (36)\quad
  \psi_\varepsilon^{i,\omega} = \pi_1(u_{i,\omega,\varepsilon} ; P^{+}),
  \quad \text{où } u_r = u_{i,\omega,\varepsilon} = u_\varepsilon^{i,\omega} :
\]

\[
  \big(U_{0,3}(\mathbf{C}), P^{+}\big) \longrightarrow (\Sigma^{*}, P_\omega)
\]

\[
  \psi_r : \Pi_{0,3} \xrightarrow{\ \sim\ } \Pi_\omega, \qquad
  r \in \operatorname{Rep}(J, \boldsymbol{\varpi})
  \simeq \operatorname{Isom}\big((\Sigma_0^{*}, P_{+}), (\Sigma^{*}, P_{+})\big)
\]

\[
  g \in \widetilde{\mathfrak{S}}_J \subset \Gamma = \check{\mathfrak{S}}_J
\]

\[
  \Gamma_{P_\varepsilon}
  = \{\, (g, \tau^{\alpha}) \in \mathfrak{S}_J \times \mathbf{Z}/2\mathbf{Z} \mid
  \alpha = 0 \text{ si } \operatorname{sg} g = 1,\
  \alpha = 1 \text{ si } \operatorname{sg} g = -1 \,\}
\]

\[
  = \{\, g \in \Gamma \mid g(\varepsilon) = \varepsilon \,\}
\]

\[
  (37)\qquad
  \pi_1(g) \text{ par transport de structure}
\]

\[
  u_r : \widetilde{\mathfrak{S}}_3 \xrightarrow{\ \sim\ } \widetilde{\mathfrak{S}}_3
\]

\[
  r' = r g_0 \quad \text{avec } g_0 \in \widetilde{\mathfrak{S}}_3
\]

\[
  (38)\quad \text{i.e. } u_{r'} = u_r u_{g_0}
  \qquad (\text{avec } u_r(g_0) = g \text{ mais peu importe})
\]

\[
  (39)\quad \psi_{r'} = \psi_r \circ \pi_1(g_0, P_{+})
  \qquad \text{si } r' = r \circ g_0
\]

\[
  g_0 = \rho \quad \text{donc} \quad
  r' = r\rho = (\omega i, \omega, \varpi) \quad \text{si } r = (i, \omega, \varpi)
\]

\[
  (40)\quad
  \psi_\varepsilon^{\omega i, \omega}
  = \overbrace{\psi_\varepsilon^{i,\omega} \circ \rho}^{\rho_\omega \circ \psi_\varepsilon^{i,\omega} =}
\]

\[
  \widetilde{\sigma}_\infty : \quad
  0 \mapsto 1, \quad 1 \mapsto 0, \quad \infty \mapsto \infty,
  \qquad \varpi \mapsto \varpi'
\]

\[
  g_0 = \widetilde{\sigma}_\infty, \qquad
  r' = r \sigma_\infty = (\omega' i, \omega', \varpi')
\]

\[
  (41)\quad
  \psi_\varepsilon^{\omega' i, \omega'}
  = \overbrace{\psi_\varepsilon^{i,\omega} \circ \widetilde{\sigma}_\infty}^{\widetilde{\sigma}_{?} \circ \psi_\varepsilon^{i,\omega} =}
\]

\[
  \psi_r = \psi_i^{\omega,\varepsilon}
  = \pi_1(\underbrace{u_{i,\omega,\varepsilon}}_{u_r} ; Q_\infty)
\]

\[
  (\infty,\ \omega_0 = +1,\ \varpi_0 = +1 \text{ donc } \varepsilon = \omega_0 \wedge \varpi_0 = +1)
\]

\[
  \psi_{r'} = \psi_r \circ \pi_1(g_0 ; Q_\infty) \qquad \text{si } r' = r \circ g_0
\]

\[
  (39)\quad
  \left\{
  \begin{array}{l}
    \psi_i^{\omega,\varepsilon'} = \tau \circ \psi_i^{\omega,\varepsilon}
      = \psi_i^{\omega,\varepsilon} \circ \tau_\infty \\[4pt]
    \psi_i^{\omega',\varepsilon'} = \sigma_i \circ \psi_i^{\omega,\varepsilon}
      = \psi_i^{\omega,\varepsilon} \circ \sigma_\infty
  \end{array}
  \right.
\]

\[
  (40)\quad
  \psi_i^{\omega',\varepsilon}
  = \widetilde{\sigma}_i \circ \psi_i^{\omega,\varepsilon}
  = \psi_i^{\omega,\varepsilon} \circ \widetilde{\sigma}_\infty
\]

\[
  \widetilde{\Sigma}_r = \widetilde{\Sigma}_i^{\omega,\varepsilon}
\]

\[
  P_\varepsilon Q_{\omega' i} R_i^{\omega} S_i^{\varepsilon}
  \simeq P_{+} Q_\infty R_0^{+} S_0^{+} .
\]

\[
  \tau(\Sigma_r), \quad \widetilde{\sigma}_i(\Sigma_r), \quad
  \widetilde{\sigma}_{\omega i}(\Sigma_r)
\]

\[
  (41)\quad
  \widetilde{\Sigma}_r = \widetilde{\Sigma}_i^{\omega,\varepsilon}
  = \widetilde{D}_r \cup \tau(\widetilde{D}_r) \cup \widetilde{\sigma}_i(\widetilde{D}_r)
  \cup \widetilde{\sigma}_{\omega i}(\Sigma_r)
\]

\[
  P_\varepsilon Q_{\omega' i} R_{\omega' i}^{\omega} S_i^{\varepsilon} \ldots
  P_{\varepsilon'} \ldots Q_i R_{\omega' i}^{\omega'} S_{\omega' i}^{\varepsilon} P_\varepsilon
\]

\[
P_+\,Q_1\,R_0^-\,S_0^+\,R_0^+\,S_0^-\,P_-\,Q_\infty\,R_1^-\,S_1^+\,P_+ .
\]

\[
\Pi_r = \Pi_i^{\omega,\varepsilon} = \pi_1(\widetilde{\Sigma};\widetilde{\Sigma}_r) = \pi_1(\Sigma^{*};\Sigma^{*}_r)
\]

\[
\text{(42)}\qquad \Psi_r : \Pi_{0,3} \longrightarrow \Pi_r = \Pi_i^{\omega,i} = \pi_1(\widetilde{\Sigma};\widetilde{\Sigma}_r)
\]

\[
\text{(43)}\qquad \Psi_i^{\omega} : \Pi_{0,3} \xrightarrow{\ \sim\ } \pi_1(\widetilde{\Sigma},R_i^{\omega})
\]

\[\text{(44)}\]

\[
\Pi_i^{\omega} = \pi_1(\widetilde{\Sigma},R^{\omega}_{\omega i}) \quad\text{à}\quad \Pi_i^{\omega'} = \pi_1(\widetilde{\Sigma},R^{\omega'}_{\omega' i}),
\]

\[
\text{(45)}\qquad \widetilde{b}_i^{\omega}\bigl(\Psi_i^{\omega}(x)\bigr) = \Psi_i^{\omega'}\bigl(\sigma_\infty(x)\bigr) .
\]

\[
\text{(1)}\qquad \Pi_1(X) \longrightarrow \widehat{\Pi}_1(X)
\]

\[
\text{(2)}\qquad
\underbrace{\operatorname{Aut}_{\Pi_1(X)}(x)}_{\overset{\text{déf}}{=}\ \pi_1(X,x)} \longrightarrow \underbrace{\operatorname{Aut}_{\widehat{\Pi}_1(X)}(x)}_{\overset{\text{déf}}{=}\ \widehat{\pi}_1(X_{\mathrm{sch}},x)}
\]

\[
\text{(3)}\qquad \pi_1(X_{\mathrm{sch}},x) \simeq \widehat{\pi}_1(X,x) .
\]

\[
\operatorname{Hom}_{\Pi_1(X)}(x,y) \longrightarrow \operatorname{Hom}_{\widehat{\Pi}_1(X)}(x,y)
\]

\[
\pi_1(X,y) \to \widehat{\pi}_1(X,y), \qquad \pi_1(X,x) \to \widehat{\pi}_1(X,x)
\]

\[
\text{(5)}\qquad \Gamma = \Gamma_K \overset{\text{déf}}{=} \operatorname{Aut}_{K\text{-cat}}(\mathbb{C}),
\]

\[
{}^{u}\lambda : {}^{u}x \longrightarrow {}^{u}y \quad\text{si}\quad \lambda : x \to y \ \ (\text{dans } \widehat{\Pi}_1(X)) .
\]

\[
\text{(6)}\qquad
\begin{cases}
u(\mathrm{id}_x) = \mathrm{id}_{{}^{u}x} & (x \in X)\\
u(\mu\lambda) = {}^{u}\mu\,{}^{u}\lambda & \Bigl(\begin{array}{l} x \xrightarrow{\lambda} y \xrightarrow{\mu} z \\ {}^{u}x \xrightarrow{{}^{u}\lambda} {}^{u}y \xrightarrow{{}^{u}\mu} {}^{u}z \end{array}\Bigr)
\end{cases}
\]

\[
\text{(7)}\qquad \overline{\Gamma} = \operatorname{Aut}_{K\text{-cat}}(\overline{K}) = \operatorname{Gal}_{\overline{K}/K} ,
\]

\[
\text{(*)}\qquad {}^{u}l : {}^{u}x = x \longrightarrow {}^{u}y = y ,
\]

\[
\text{(8)}\qquad
\begin{cases}
{}^{u}l = g_l(u)\circ l = l \circ f_l(u) \\
g_l(u) \in \widehat{\pi}_1(X,y) \\
f_l(u) \in \widehat{\pi}_1(X,x)
\end{cases}
\]

\[
\text{(9)}\qquad
\begin{cases}
f_l(u) = l^{-1}\cdot{}^{u}l , & g_l(u) = {}^{u}l\cdot l^{-1} \\
g_l(u) = l\bigl(f_l(u)\bigr) , & f_l(u) = l^{-1}\bigl(g_l(u)\bigr)
\end{cases}
\]

\[
x \xrightarrow{\ l'\ } y \xrightarrow{\ l\ } z
\]

\[
\text{(10)}\qquad g_{ll'}(u) = g_l(u)\cdot l\bigl(g_{l'}(u)\bigr) \quad\text{soit}\quad g_u(ll') = g_u(l)\cdot l\bigl(g_u(l')\bigr)
\]

\[
\text{(11)}\qquad f_{ll'}(u) = l'^{-1}\bigl(f_l(u)\bigr)\cdot f_{l'}(u) \quad\text{i.e.}\quad f_u(ll') = l'^{-1}f_u(l)\cdot f_u(l')
\]

\[
\text{(12)}\qquad \text{\struck{$g_l : \Gamma \longrightarrow \widehat{\pi}_1(X,y)$}}
\]

\[
x_0 \xrightarrow{\ l_1\ } x_1 \xrightarrow{\ l_2\ } \cdots \; x_{n-1} \xrightarrow{\ l_n\ } x_n
\]

\[
\text{(12)}\qquad g_{l_n l_{n-1}\cdots l_1}(u) = g_{l_n}(u)\cdot l_n\bigl(g_{l_{n-1}}(u)\bigr)\cdot l_n l_{n-1}\bigl(g_{l_{n-2}}(u)\bigr)\cdots
\]

\[
\cdots (l_n\cdots l_2)\,g_{l_1}(u) .
\]

\[
\text{(13)}\qquad p : T \longrightarrow X
\]

\[
\text{(14)}\qquad \text{\struck{$\Pi_1(X;$}}\ \pi_1(X;T) = \pi
\]

\[
\text{(15)}\qquad \varphi_x : \pi \xrightarrow{\ \sim\ } \pi_1\bigl(X,p(x)\bigr)
\]

\[
\text{(16)}\qquad l_{y,x} : p(x) \longrightarrow p(y) \quad\text{dans } \Pi_1(X)
\]

\[
\text{(17)}\qquad g^{T/X}_{y,x}(u) = g^{p}_{y,x}(u) = \varphi_y^{-1}\bigl(g_{l_{y,x}}(u)\bigr) = \varphi_x^{-1}\bigl(f_{l_{y,x}}(u)\bigr) \qquad \bigl(\in \widehat{\pi},\ \pi = \pi_1(X;T)\bigr) .
\]

\[
\text{(18)}\qquad g_{zy}(u)\,g_{yx}(u) = g_{zx}(u) \qquad x,y,z \in T_K
\]

\[
\text{(19)}\qquad g_{x_n,x_0} = g_{x_n,x_{n-1}}\,g_{x_{n-1},x_{n-2}}\cdots g_{x_1,x_0}
\]

\[
\text{(20)}\qquad f_l(u) = \varphi_x\bigl(g^{T}_{y_0,x_0}(u)\bigr), \qquad g_l(u) = \varphi_y\bigl(g^{T}_{y_0,x_0}(u)\bigr)
\]

\[
\text{\struck{$x_0 = 0 \in T$, $y_0$}} \quad x_0 = 0,\ y_0 = 1 \in T = [0,1] .
\]

\[\text{(21)}\]

\[
f_K : T_K \longrightarrow T'_K ,
\]

\[
\text{(22)}\qquad \pi = \pi_1(X;T/X) \xrightarrow{\ \psi = \pi_1(f)\ } \pi' = \pi_1(X,T'/X)
\]

\[
\text{(23)}\qquad g^{T'}_{y',x'}(u) = \psi\bigl(g^{T}_{y,x}(u)\bigr)
\]

\[
\bigl(x' = f(x),\ y' = f(y),\ \psi = \pi_1(f)\bigr)
\]

\[
\text{\struck{$g^{\alpha} : T_{\alpha K} \times T_{\alpha K} \to$}}
\]

\[
\text{(24)}\qquad \pi_\alpha \overset{\text{déf}}{=} \pi_1(X, T_\alpha)
\]

\[
\text{(25)}\qquad
\left\{
\begin{array}{l}
T_{\alpha K} \times T_{\alpha K} \times \Gamma \longrightarrow \widehat{\pi}_\alpha \\
(x,y,u) \longmapsto g^{T_\alpha}_{y,x}(u) \overset{\text{déf}}{=} g^{\alpha}_{y,x}(u)
\end{array}
\right.
\]

\[
\text{(26)}\qquad \text{\struck{$\ill{}$}}\ \ \varphi_{\beta\alpha} : \pi_\beta \longrightarrow \pi_\alpha
\]

\[
\text{(27)}\qquad g^{\beta}_{y,x}(u) = \varphi_{\beta\alpha}\bigl(g^{\alpha}_{y,x}(u)\bigr) \qquad x, y \in T_{\alpha K},\ u \in \Gamma
\]

\[
\text{(28)}\qquad \forall \alpha, \beta \in I,\ \ T_\alpha \cap T_\beta = \bigcup_{\gamma \leq \alpha,\beta} T_\gamma
\]

\[
\text{\struck{$i_0 = i,\ i_1, \ldots, i_n = j$}}
\]

\[
g^{\alpha}_{y,x}(u) \in \widehat{\pi}_\alpha \qquad (\alpha \in A,\ y, x \in T_\alpha)
\]

\[
I_\alpha \subset T_\alpha \quad \text{partie de } T_\alpha \ (\text{\uncertain{non vide}, \uncertain{finie}}),
\]

\[
\text{(28)}\qquad {}^{uv}l = {}^{u}({}^{v}l) \qquad u, v \in \Gamma,\ l : x \to y \ \text{chemin dans } \Pi_1(X)
\]

\[
{}^{v}l = g_l(v)\cdot l
\]

\[
{}^{u}({}^{v}l) = {}^{u}\{g_l(v)\}\cdot\underbrace{{}^{u}l}_{g_l(u)\cdot l} = {}^{u}g_l(v)\,g_l(u)\cdot l
\]

\[
\text{(29)}\qquad
\left|\begin{array}{l}
g_l(uv) = {}^{u}g_l(v)\cdot g_l(u) \quad\text{soit}\\
g_l^{-1}(uv) = g_l^{-1}(u)\cdot{}^{u}g_l^{-1}(v)
\end{array}\right.
\]

\[
\text{(30)}\qquad f_l(uv) = f_l(u)\cdot{}^{u}f_l(v)
\]

\[
\widehat{\pi}_1(X,x) \xrightarrow{\ \pi_1(l)\ } \widehat{\pi}_1(X,y) \quad \text{induit par } l \ldots
\]

\[
l' = la = bl \qquad a \in \pi_1(X,x),\ b \in \pi_1(X,y)
\]

\[
\text{(31)}\qquad
\left\{
\begin{array}{l}
f_{l'}(u) = \operatorname{int}(a^{-1})\,f_l(u)\,\underbrace{f_a(u)}_{a^{-1}\cdot{}^{u}a} = a^{-1}\cdot f_l(u)\cdot{}^{u}a \\[1ex]
g_{l'}(u) = \underbrace{g_b(u)}_{{}^{u}b\cdot b^{-1}}\,\operatorname{int}(b)\,g_l(u) = {}^{u}b\,g_l(u)\,b^{-1}
\end{array}
\right.
\]

\[
\bigl(\text{i.e.}\ \ g^{-1}_{l'}(u) = b\,g^{-1}_l(u)\,{}^{u}b^{-1}\bigr)
\]

\[
\text{(32)}\qquad
\left\{
\begin{array}{l}
g_{y,x} \in H^1\bigl(\Gamma, \widehat{\pi}_1(X,y)\bigr) \\
f_{y,x} \in H^1_{\mathrm{dr}}\bigl(\Gamma, \widehat{\pi}_1(X,x)\bigr)
\end{array}
\right.
\]

\[
\widehat{\pi}_1(X,x) \xrightarrow[\text{ext}]{\ \text{isom}\ } \widehat{\pi}_1(Y,y)
\]

\[
\text{(33)}\qquad
\left\{
\begin{array}{l}
f_l : \Gamma \longrightarrow \widehat{\pi}_1(X,x) \\
g_l : \Gamma \longrightarrow \widehat{\pi}_1(Y,y)
\end{array}
\right.
\qquad l : x \to y \ \ (\text{dans } \Pi_1(X))
\]

\[
f_{x_0,x} \in H^1_{\mathrm{dr}}\bigl(\Gamma, \underbrace{\widehat{\pi}_1(X,x_0)}_{\widehat{\pi}(x_0)}\bigr),
\]

\[
f_l : \Gamma \longrightarrow \widehat{\pi}(x_0)
\]

\[
  (34) \qquad (f_{x_0, x})_{\mathrm{ab}} \in H^1\bigl(\Gamma, \hat{\pi}_1(X, x_0)_{\mathrm{ab}}\bigr),
\]

\[
  (35) \qquad U_{0,3} = \mathbb{P}^1 \setminus \{0, 1, \infty\}.
\]

\[
  (36) \qquad f : X \to Y ;
\]

\[
  (37) \qquad \Pi_1(X) \xrightarrow{\; f_{*} = \Pi_1(f) \;} \Pi_1(Y),
\]

\[
  (38) \qquad \hat{\Pi}_1(X) \xrightarrow{\; \hat{f} = \hat{\Pi}_1(f) \;} \hat{\Pi}_1(Y),
\]

\[
  (39) \qquad \hat{f}({}^{u}\ell) = {}^{u}(\hat{f}\ell)
\]

\[
  (40) \qquad
  \begin{cases}
    \hat{f}\bigl(f_{\ell}(u)\bigr) = f_{\hat{f}(\ell)}(u) \\
    \hat{f}\bigl(g_{\ell}(u)\bigr) = g_{\hat{f}(\ell)}(u) .
  \end{cases}
\]

\[
  (41) \qquad \boxed{\mathrm{Hom}_{\hat{\Pi}_1(X, I)}(x_0, x_i)}, \qquad i \in I,
\]

\[
  (42) \qquad f_{\ell_i}(u), \qquad i \in I,\ u \in \Gamma,
\]

\[
  (43) \qquad f_{\lambda_j}(u) \ \text{ou}\ g_{\lambda_j}(u), \qquad j \in J,\ u \in \Gamma .
\]

\[
  \mathrm{Hom}_{\hat{\Pi}_1(C)}(x, y) \longrightarrow \mathrm{Hom}_{\hat{\Pi}_1(U_{0,3})}(x, y)
\]

\[
  \hat{\Pi}_1(C, I) \longrightarrow \hat{\Pi}_1(X, I),
\]

\[
  \hat{\pi}_1(C, x_0) \subset \hat{\pi}_1\bigl(U_{0,3}, f(x_0)\bigr)
\]

\[
  \hat{\pi}_1(C, x_0) \longrightarrow \hat{\pi}_1(X, x_0) ;
\]

\[
  (45) \qquad \mathcal{C}' \longrightarrow \tilde{\mathcal{C}}'
\]

\[
  \pi_1(\tilde{\mathcal{C}}, x_0') \hookrightarrow \pi_1(\mathcal{C}, x_0) = \Pi_0,
\]

\[
  (45) \qquad \mathrm{Ob}\,\tilde{\mathcal{C}}_{x_i} = H_{x_i, x_0} \wedge^{\Pi_0} \Pi_0/\Pi_0'
\]

\[
  (46) \qquad \tilde{\Theta}_0(g \cdot x') = \Theta(g)\, \tilde{\Theta}_0(x')
\]

\[
  (47) \qquad E_0 \simeq \Pi_0/\Pi_0' ;
\]

\[
  (48) \qquad \mathrm{int}\, g\,(\Pi_0') = \Theta(\Pi_0'),
\]

\[
  (49) \qquad \mathrm{Transp}_{\Pi_0}\bigl(\Pi_0', \Theta(\Pi_0')\bigr)/\Pi_0'
\]

\[
  \mathcal{C} = \hat{\Pi}_1(X_{\bar K}, I), \qquad I \subset X_K,
\]

\[
    (*) \qquad \hat{\pi}_1(X', x') \longrightarrow \hat{\pi}_1\bigl(Y, f(x')\bigr)
  \]

\[
  (50) \qquad
  \begin{aligned}
    E_0 &\longrightarrow \text{Ens.\ des s-groupes de } \Pi, \\
    x' &\longmapsto \text{stabilisateur } \Pi_{x'},
  \end{aligned}
\]

\[
  (51) \qquad E_0/\Pi_0 \overset{\mathrm{def}}{=} \pi_0(E_0) \longrightarrow \Sigma = \mathrm{Ssgr}(\Pi)/\Pi
\]

\[
  (52) \qquad \pi_0(\tilde{\Theta}_0) : \pi_0(E_0) \xrightarrow{\;\sim\;} \pi_0(E_0)
\]

\[
  \hat{\pi}_1(X', x) \longrightarrow \hat{\pi}_1\bigl(Y, f(x)\bigr) .
\]

\[
  \text{\struck{$I'_1 \cap f^{-1}(I'_2) \longrightarrow$}} \quad
  \hat{\Pi}_1(X'_1, J) \longrightarrow \hat{\Pi}_1(X'_2, I'_2)
\]

\[
  X'_1 \xrightarrow{\;f_1\;} Y, \qquad X'_2 \xrightarrow{\;f_2\;} Y
\]

\begin{equation*}
  \tag{53} u\, l^{T}_{y';x'} \;=\; g^{T}_{x,y}(u)\; l^{T}_{u(y'),u(x')}
\end{equation*}

\[
  g^{T}_{x',y'}(u) \in \hat\pi_1\bigl(X',\, T(u(x'),u(y'))\bigr)
\]

\begin{equation*}
  \tag{54} \hat f\bigl(\underbrace{g^{T}_{y';x'}(u)}_{\in\, \hat\pi_1(X,T)}\bigr) \;=\; g^{T}_{y,x}(u)
\end{equation*}

\[
  \hat\varphi_\alpha : \hat\pi_1(X'_{\bar K}, T_\alpha) \longrightarrow \hat\pi_1(X, T)
\]

\[
  g^{T}_{y,x}(u) \in \bigcap_{\alpha \in \pi_0(X'_T)} \operatorname{Im} \hat\varphi_\alpha
  \qquad \forall u \in \Gamma = \Gamma_{\bar K/K}
\]

\[
  z \mapsto -z, \qquad z \mapsto \frac{1}{z}, \qquad \text{\struck{\ill{}}}
\]

\[
  1 \longrightarrow \mathbb{F}_2^{\,2} \longrightarrow \mathfrak{A}_4 \longrightarrow \mathbb{Z}/3\mathbb{Z} \longrightarrow 1
\]

\[
  l_\infty\, l_1\, l_0\, l_{-1} = 1
\]

\[
  E_0 \;\text{---}\; \pi_0(E_0) \;\text{\struck{$\simeq E_0/\pi \;\mathrm{Ssgr}($}}
\]

\[
  \Sigma \;(= \Gamma \cdot \pi) \supset \pi \;\text{---}\; E_0 \longrightarrow \mathrm{Ssgr}(\pi)/\pi,
  \qquad \Sigma/\pi = \Gamma
\]

\[
  \Sigma_J = \mathbb{P}\bigl(V(J)\bigr) \simeq \check{\mathbb{P}}\bigl(V(J)\bigr)
\]

\[
  F(1) \simeq \mathcal{O}(\omega)
\]

\[
  \check{\mathcal{O}}(1) \simeq \check F(\omega)
\]

\[
  \Omega^1_\Sigma \simeq F(-1) \simeq \mathcal{O}(-2)(\omega), \qquad
  \check\Omega^1_\Sigma \simeq \mathcal{O}(2)(\omega) \quad \text{fibré tangent à } \Sigma
\]

\[
  \Gamma\bigl(\Sigma_J, \mathcal{O}_\Sigma(1)\bigr) \simeq V(J)
  \qquad \text{\struck{$e_j - e_k$ section $\omega_{j,k}$}}
\]

\[
  \varphi_{jk} = e_k - e_j \in \Gamma\bigl(\Sigma, \mathcal{O}_\Sigma(1)\bigr) \quad \text{nulle en } R_i,
  \qquad \varphi_{kj} = -\varphi_{jk}
\]

\[
  \varphi_{jk}(R_i) = 0 \qquad
  \boxed{\varphi_{jk}(R_k) = \tilde R_k, \quad \varphi_{jk}(R_j) = -\tilde R_j}
\]

\[
  \tilde Q_i = \varphi_{jk}(Q_i), \qquad \tilde Q'_i = -\varphi_{jk}(Q_i) = \varphi_{kj}(Q_i)
\]

\[
  \boxed{Q^\omega_{i} = \varphi_{\omega i, \omega' i}(Q_i)}
\]

\[
  \Sigma_n^{\ast\ast} = \Sigma_n \setminus \bigl\{ S(\Sigma_n) \cup A(\Sigma_n) \cup F(\Sigma_n) \bigr\}
\]

\[
  \Sigma_n^{\ast\ast} \xrightarrow{\ \deg n\ } \Sigma_1^{\ast\ast} = \mathbb{P}_1 \setminus \{\underbrace{0, \infty}_{\text{« pôles »}}, 1, -1\}
  \xrightarrow{\ \deg 2\ } \Sigma_3^{\ast\cdot} = U_{0,3}
\]

\[
  \Sigma_n^{\ast\ast} \simeq \Sigma_1^{\ast\ast}/D_n \;\; \text{\uncertain{(sic)}}
\]

\[
  \Sigma_{2n}^{\ast\cdot} \longrightarrow \Sigma_1^{\ast\cdot} \xrightarrow{\ 2\ } \Sigma_1^{\ast\cdot} \simeq U_0^3
\]

\[
  \Sigma_n^{\ast\ast} \xrightarrow{\ D_n^+\ } \Sigma_1^{\ast\cdot} \simeq U_0^3
\]

\[
  Q^\omega_i = R_j - R_k \longmapsto R_j - \zeta R_k = P^\omega_i \;\text{\struck{\ill{}}}\; = P^{\omega,\zeta}_i
\]

\[
  P^{\omega,\zeta}_i = R_{\omega i} \;\text{\struck{$\zeta$}}\; - \zeta R_{\omega^2 i} \;\equiv\; -P^{\omega,\zeta}_{\omega^2 i}
\]

\[
  P^{\omega',\zeta'}_i = R_{\omega' i} - \zeta' R_{\omega i} = R_k - \zeta^2 R_j = -\zeta' P^{\omega,\zeta}_i
\]

\[
  S = S_0 \sqcup S_1 \sqcup S_\infty \subset \mathcal{X}
\]

\[
  \tau_0(r),\ \tau_1(r),\ \tau_\infty(r),
\]

\begin{equation*}
  \tag{1}
  \begin{cases}
    \tau_1(\tau_0(r)) = \rho_\infty(r) \\
    \tau_0(\tau_1(r)) = \rho_\infty^{-1}(r)
  \end{cases}
  \qquad \text{(face conservée)}
\end{equation*}

\begin{equation*}
  \tag{2}
  \begin{cases}
    \tau_\infty \tau_1(r) = \rho_0(r) \\
    \tau_1 \tau_\infty(r) = \rho_0^{-1}(r)
  \end{cases}
  \qquad \text{(sommet conservé)}
\end{equation*}

\begin{equation*}
  \tag{3}
  \tau_0 \tau_\infty(r) = \tau_\infty \tau_0(r) \overset{\text{def}}{=} \rho_1(r)
\end{equation*}

\begin{equation*}
  \tag{4}
  \tau_0^2 = \tau_1^2 = \tau_\infty^2 = (\tau_0 \tau_\infty)^2 = 1
\end{equation*}

\begin{equation*}
  \tag{5}
  \begin{cases}
    \rho_\infty = \tau_1 \tau_0 \\
    \rho_0 = \tau_\infty \tau_1 \\
    \rho_1 = \tau_0 \tau_\infty
  \end{cases}
  \qquad \rho_\infty \rho_1 \rho_0 \;\text{\struck{$\rho_\infty$}}\; = 1
\end{equation*}

\begin{equation*}
  \tag{6} R^{i,j}_r \qquad r \in \mathcal{R},\ (i,j) \in \mathrm{Rep}\{0,1,\infty\}
  \quad \text{i.e.\ } i,j \in \{0,1,\infty\},\ i \neq j
\end{equation*}

\begin{equation*}
  \tag{7} \boxed{R^{i,j}_r = R^{i,j}_{\tau_k(r)}, \quad k \notin \{i,j\}}
\end{equation*}

\begin{equation*}
  \tag{8} \lambda^i_r : R^{i,i+1}_r \longrightarrow R^{i,i+2}_r \qquad r \in \mathcal{R},\ i \in \{0,1,\infty\}
\end{equation*}

\begin{equation*}
  \tag{9} a^i_r : R^{i+1,i}_r \;\text{\struck{\ill{}}}\; \longrightarrow R^{i+1,i+2}_r \qquad r \in \mathcal{R},\ i \in \{0,1,\infty\}
\end{equation*}

\begin{equation*}
  \tag{10} \boxed{a^i_r = a^i_{\tau_i(r)}}
\end{equation*}

\begin{equation*}
  \tag{11} \tilde X^{\ast\ast} \longrightarrow X
\end{equation*}

\begin{equation*}
  \tag{12}
  \boxed{\underbrace{(\lambda^0_r)^{-1} a^1_r}_{\overset{\text{def}}{=}\, b^\infty_r}\;
  \underbrace{(\lambda^\infty_r)^{-1} a^0_r}_{b^1_r}\;
  \underbrace{(\lambda^1_r)^{-1} a^\infty_r}_{b^0_r} = 1}
  \qquad \text{lacet en } R^{1\infty}_r
\end{equation*}

\[
  \text{\struck{\ill{}}} \qquad (\tau_0 \tau_\infty)^2 = 1 \quad \text{i.e.} \quad \rho_1^2 = 1
\]

\[
  \tau_0 \tau_\infty(r) = \tau_\infty \tau_0(r) \qquad \text{pour tout } r.
\]

\begin{equation*}
  \tag{13}
  \begin{cases}
    \Lambda^i_r = (\lambda^i_{\tau_{i+1}(r)})^{-1} \lambda^i_r : R^{i,i+1}_r \longrightarrow R^{i,i+1}_{\tau_{i+1}(r)} \\
    \Lambda'^i_r = \lambda^i_{\tau_{i-1}(r)} (\lambda^i_r)^{-1} : R^{i,i-1}_r \longrightarrow R^{i,i-1}_{\tau_{i-1}(r)}
  \end{cases}
\end{equation*}

\begin{equation*}
  \tag{14}
  \boxed{\Lambda^i_r \Lambda^i_{\tau_{i+1}(r)} = 1} \qquad
  \boxed{\Lambda'^i_r \Lambda'^i_{\tau_{i-1}(r)} = 1}
\end{equation*}

\begin{equation*}
  \tag{14} \{R^{ij}_r\} \quad
  \begin{cases}
    r \in \mathcal{R} \\
    i \text{ ou } j \text{ est égal à } 1 \text{ i.e.\ } \{i,j\} \neq \{0,\infty\}
  \end{cases}
\end{equation*}

\begin{equation*}
  \tag{15}
  \begin{cases}
    \lambda^1_r & r \in \mathcal{R} \\
    \Lambda^0_r,\ \Lambda^\infty_r & r \in \mathcal{R},\ \text{\struck{\ill{}}} \\
    a^i_r & r \in \mathcal{R},\ i \in \{0, \infty\}
  \end{cases}
  \qquad \Lambda^0_r \Lambda^0_{\tau_1 r} =
\end{equation*}

\begin{equation*}
  \tag{16}
  \text{\struck{$\Lambda^\infty_{\tau_1 r} (a_{\tau_1 r})^{-1} \lambda^1_{\tau_1 r} (a^\infty_{\tau_1 r})^{-1} \Lambda^0_r a^\infty_r \lambda^1_r a^0_r = 1$}}
\end{equation*}

\begin{equation*}
  \tag{16}
  \boxed{\Lambda'^\infty_{\tau_1(r)}
  \underbrace{\bigl(a^0_{\tau_1 r} (\lambda^1_{\tau_1 r})^{-1} a^\infty_{\tau_1 r}\bigr)}_{\beta^1_{\tau_1(r)}}{}^{\!-1}
  \Lambda^0_r
  \underbrace{\bigl((a^\infty_r)^{-1} \lambda^1_r (a^0_r)^{-1}\bigr)}_{\beta^{1\,-1}_r} = 1}
\end{equation*}

\begin{equation*}
  \tag{17} R^{ij}_r \quad
  \begin{cases}
    r \in \mathcal{R} \\
    \{i, j\} = \{0, 1\}
  \end{cases}
  \qquad\quad
  \boxed{
  \begin{cases}
    R^{0,1}_r = R^{0,1}_{\tau_\infty r} \\
    R^{1,0}_r = R^{1,0}_{\tau_\infty(r)}
  \end{cases}}
\end{equation*}

\begin{equation*}
  \tag{18}
  \begin{cases}
    a^\infty_r : R^{0,1}_r \;\text{\struck{\ill{}}}\; \longrightarrow R^{1,0}_r & (r \in \mathcal{R}) \\
    \Lambda^0_r \;\text{\struck{$\Lambda^1_r$}} : R^{01}_r \longrightarrow R^{01}_{\tau_1(r)} \\
    \Lambda'^1_r : R^{10}_r \longrightarrow R^{1,0}_{\tau_0(r)}
  \end{cases}
\end{equation*}

\begin{equation*}
  \tag{19}
  \boxed{
  \begin{aligned}
    \Lambda^0_r \Lambda^0_{\tau_1(r)} &= 1 \\
    \Lambda'^1_r \Lambda'^1_{\tau_0(r)} &= 1
  \end{aligned}}
\end{equation*}

\begin{equation*}
  \tag{20} R^{0,1}_r \qquad r \in \mathcal{R}
  \qquad\quad
  \boxed{R^{0,1}_r = R^{0,1}_{\tau_\infty r}}
\end{equation*}

\begin{equation*}
  \tag{21}
  \begin{cases}
    A^{0,1}_r \overset{\text{def}}{=} (a^\infty_{\tau_0 r})^{-1} \;\text{\struck{\ill{}}}\; (\Lambda'^1_r)^{-1} a^\infty_r : R^{0,1}_r \longrightarrow R^{0,1}_{\tau_0(r)} \\
    \Lambda^0_r : R^{0,1}_r \longrightarrow R^{0,1}_{\tau_1 r} & (r \in \mathcal{R})
  \end{cases}
\end{equation*}

\begin{equation*}
  \tag{22}
  \boxed{
  \begin{cases}
    \Lambda^0_r \Lambda^0_{\tau_1(r)} = 1 & r \in \mathcal{R} \\
    A^{0,1}_r A^{0,1}_{\tau_0(r)} = 1 & r \in \mathcal{R}
  \end{cases}}
\end{equation*}