Cote n° 146 · pages 2–132
· 359 displayed formulas · Groupoïdes de Teichmüller et les S0 Tg,!ν, S0 TG : notes manuscrites (s.d.).
Inventory dating : [à partir de 1978]
Édition de démonstration
\[\mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathbb{Z}^{\hat A}\]
LaTeX source
\[
\mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathbb{Z}^{\hat A}
\]\[(1) \qquad S_0\mathcal{T}_G = S_0\mathcal{T}_{G^b} \wedge^{\mathbb{Z}^{\vec{\hat A}}} \mathbb{Z}^{\hat A}\]
LaTeX source
\[
(1) \qquad S_0\mathcal{T}_G = S_0\mathcal{T}_{G^b} \wedge^{\mathbb{Z}^{\vec{\hat A}}} \mathbb{Z}^{\hat A}
\]\[(2) \qquad S_0\mathcal{T}_{G^b} = \prod_{s\in S} S_0\mathcal{T}_{G^b_s} = \prod_{s\in S} S_0\mathcal{T}_{0,\vec{\hat A}_s} ,\]
LaTeX source
\[
(2) \qquad S_0\mathcal{T}_{G^b} = \prod_{s\in S} S_0\mathcal{T}_{G^b_s} = \prod_{s\in S} S_0\mathcal{T}_{0,\vec{\hat A}_s} ,
\]\[T_i \xleftrightarrow[\ \sim\ ]{\ \varphi_{\alpha,i}\ } \mathbb{U}
\qquad \mathbb{U} = \{ z \in \mathbb{C} \mid |z| = 1 \}\]
LaTeX source
\[
T_i \xleftrightarrow[\ \sim\ ]{\ \varphi_{\alpha,i}\ } \mathbb{U}
\qquad \mathbb{U} = \{ z \in \mathbb{C} \mid |z| = 1 \}
\]\[\mathbb{U}^I \xrightarrow[\ \sim\ ]{\ \varphi\ } P\mathbb{U}_G = P\mathbb{U}_{0,I} = \prod_{i\in I} T_i\]
LaTeX source
\[
\mathbb{U}^I \xrightarrow[\ \sim\ ]{\ \varphi\ } P\mathbb{U}_G = P\mathbb{U}_{0,I} = \prod_{i\in I} T_i
\]\[(3) \qquad \omega(I) \hookrightarrow P\mathbb{U}_{0,I}\]
LaTeX source
\[
(3) \qquad \omega(I) \hookrightarrow P\mathbb{U}_{0,I}
\]\[(5) \qquad 0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{U} \to 0 ,
\qquad t \mapsto \exp 2i\pi t\]
LaTeX source
\[
(5) \qquad 0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{U} \to 0 ,
\qquad t \mapsto \exp 2i\pi t
\]\[(5_I) \qquad 0 \to \mathbb{Z}^I \to \mathbb{R}^I \to \mathbb{U}^I \to 0 ,\]
LaTeX source
\[
(5_I) \qquad 0 \to \mathbb{Z}^I \to \mathbb{R}^I \to \mathbb{U}^I \to 0 ,
\]\[(6) \qquad 0 \to \pi \to \tilde T \to T \to 0\]
LaTeX source
\[ (6) \qquad 0 \to \pi \to \tilde T \to T \to 0 \]
\[(7) \qquad \Pi(x,y) = \tilde{\mathbb{U}}_{y-x}
\qquad \text{(fibre de $\tilde{\mathbb{U}}$ en $y - x \in \mathbb{U}$)}\]
LaTeX source
\[
(7) \qquad \Pi(x,y) = \tilde{\mathbb{U}}_{y-x}
\qquad \text{(fibre de $\tilde{\mathbb{U}}$ en $y - x \in \mathbb{U}$)}
\]\[(8) \qquad 0 \to \pi \to \mathcal{E} \to \Gamma \to 0\]
LaTeX source
\[
(8) \qquad 0 \to \pi \to \mathcal{E} \to \Gamma \to 0
\]\[(9) \qquad \mathrm{Hom}(x,y) = \mathcal{E}_{y-x}\]
LaTeX source
\[
(9) \qquad \mathrm{Hom}(x,y) = \mathcal{E}_{y-x}
\]\[(10) \qquad \mathrm{Fl}\,\mathcal{E}(\varepsilon) = \mathcal{E} \times \varepsilon\]
LaTeX source
\[
(10) \qquad \mathrm{Fl}\,\mathcal{E}(\varepsilon) = \mathcal{E} \times \varepsilon
\]\[(u, vx) \circ (v, x) = (uv, x)\]
LaTeX source
\[ (u, vx) \circ (v, x) = (uv, x) \]
\[\varepsilon' \longrightarrow \varepsilon\]
LaTeX source
\[ \varepsilon' \longrightarrow \varepsilon \]
\[\Gamma' \longrightarrow \Gamma\]
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\[ \Gamma' \longrightarrow \Gamma \]
\[(11) \qquad 0 \to \pi \to \mathcal{E}' \to \Gamma' \to 0\]
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\[
(11) \qquad 0 \to \pi \to \mathcal{E}' \to \Gamma' \to 0
\]\[(12) \qquad S_0\mathcal{T}_{0,I} \simeq \mathcal{E}_I(\omega(I))\]
LaTeX source
\[
(12) \qquad S_0\mathcal{T}_{0,I} \simeq \mathcal{E}_I(\omega(I))
\]\[(13) \qquad 0 \to \mathbb{Z}^I \to \mathcal{E}_I \to \{\pm 1\} \to 0\]
LaTeX source
\[
(13) \qquad 0 \to \mathbb{Z}^I \to \mathcal{E}_I \to \{\pm 1\} \to 0
\]\[\mu_2 \xrightarrow{\ \mathrm{diag}\ } \mu_2^I \longrightarrow \mathbb{U}^I .\]
LaTeX source
\[
\mu_2 \xrightarrow{\ \mathrm{diag}\ } \mu_2^I \longrightarrow \mathbb{U}^I .
\]\[(14) \qquad 0 \to \mathbb{Z} \xrightarrow{\ 2\ } \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 ,
\qquad \mathbb{Z}/2\mathbb{Z} \simeq \mu_2 .\]
LaTeX source
\[
(14) \qquad 0 \to \mathbb{Z} \xrightarrow{\ 2\ } \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 ,
\qquad \mathbb{Z}/2\mathbb{Z} \simeq \mu_2 .
\]\[0 \to \mathbb{Z}^I \xrightarrow{\ 2\ } \mathbb{Z}^I \to (\mathbb{Z}/2\mathbb{Z})^I \to 0 ,
\qquad (\mathbb{Z}/2\mathbb{Z})^I \simeq \mu_2^I ,\]
LaTeX source
\[
0 \to \mathbb{Z}^I \xrightarrow{\ 2\ } \mathbb{Z}^I \to (\mathbb{Z}/2\mathbb{Z})^I \to 0 ,
\qquad (\mathbb{Z}/2\mathbb{Z})^I \simeq \mu_2^I ,
\]\[(16) \qquad \underbrace{\mathrm{Hom}}_{\mathcal{E}_I(\varepsilon)}(x,y)
= T_{x-y} \wedge^{\mathbb{Z}} \mathbb{Z}^I \simeq
\begin{cases}
\mathbb{Z}^I & \text{si } x = y \\
\mathrm{Imp}^I & \text{si } x \neq y
\end{cases}\]
LaTeX source
\[
(16) \qquad \underbrace{\mathrm{Hom}}_{\mathcal{E}_I(\varepsilon)}(x,y)
= T_{x-y} \wedge^{\mathbb{Z}} \mathbb{Z}^I \simeq
\begin{cases}
\mathbb{Z}^I & \text{si } x = y \\
\mathrm{Imp}^I & \text{si } x \neq y
\end{cases}
\]\[(17) \qquad \pi_1(S_0\mathcal{T}_{0,I}, \mathfrak{S}_I; x)
= \bigl\{ \alpha, l \bigm| \alpha \in \mathfrak{S}_I,\ l : x \leftarrow \alpha x \text{ dans } S_0\mathcal{T}_{0,I} \bigr\}\]
LaTeX source
\[
(17) \qquad \pi_1(S_0\mathcal{T}_{0,I}, \mathfrak{S}_I; x)
= \bigl\{ \alpha, l \bigm| \alpha \in \mathfrak{S}_I,\ l : x \leftarrow \alpha x \text{ dans } S_0\mathcal{T}_{0,I} \bigr\}
\]\[= \bigl\{ (\alpha, (n_i)_{i\in I}) \in \mathfrak{S}_I \times \mathbb{Z}^I \bigm|
\text{les $n_i$ ont la parité définie par $\mathrm{sg}(\alpha)$} \bigr\}\]
LaTeX source
\[
= \bigl\{ (\alpha, (n_i)_{i\in I}) \in \mathfrak{S}_I \times \mathbb{Z}^I \bigm|
\text{les $n_i$ ont la parité définie par $\mathrm{sg}(\alpha)$} \bigr\}
\]\[(18) \qquad (l, \alpha)(l', \alpha') = (\underbrace{l \circ \alpha(l')}_{l + \alpha(l')}, \alpha\alpha')\]
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\[
(18) \qquad (l, \alpha)(l', \alpha') = (\underbrace{l \circ \alpha(l')}_{l + \alpha(l')}, \alpha\alpha')
\]\[0 \to \mathbb{Z} \xrightarrow{\ 2\ } \mathbb{Z} \to \underset{\mu_2}{\mathbb{Z}/2\mathbb{Z}} \to 0 ,\]
LaTeX source
\[
0 \to \mathbb{Z} \xrightarrow{\ 2\ } \mathbb{Z} \to \underset{\mu_2}{\mathbb{Z}/2\mathbb{Z}} \to 0 ,
\]\[\mathfrak{S}_I \xrightarrow{\ \mathrm{sgn}\ } \mu_2 \simeq \mathbb{Z}/2\mathbb{Z} ,\]
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\[
\mathfrak{S}_I \xrightarrow{\ \mathrm{sgn}\ } \mu_2 \simeq \mathbb{Z}/2\mathbb{Z} ,
\]\[(19) \qquad \alpha \longmapsto (0, \alpha) \qquad \alpha \in \mathfrak{S}_I^+ .\]
LaTeX source
\[
(19) \qquad \alpha \longmapsto (0, \alpha) \qquad \alpha \in \mathfrak{S}_I^+ .
\]\[S_0\mathcal{T}_{G^b} = \prod_{s\in S} S_0\mathcal{T}_{G^b_s} = \prod_{s\in S} S_0\mathcal{T}_{0,\vec{\hat A}_s}\]
LaTeX source
\[
S_0\mathcal{T}_{G^b} = \prod_{s\in S} S_0\mathcal{T}_{G^b_s} = \prod_{s\in S} S_0\mathcal{T}_{0,\vec{\hat A}_s}
\]\[(20) \qquad \mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathbb{Z}^{\hat A} .\]
LaTeX source
\[
(20) \qquad \mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathbb{Z}^{\hat A} .
\]\[\varepsilon(G) = \varepsilon(G^b) = \prod_{s\in S} \underbrace{\omega(\vec{\hat A}_s)}_{\varepsilon(G^b_s)}\]
LaTeX source
\[
\varepsilon(G) = \varepsilon(G^b) = \prod_{s\in S} \underbrace{\omega(\vec{\hat A}_s)}_{\varepsilon(G^b_s)}
\]\[(22) \qquad \gamma(G) = \gamma(G^b) = \{\pm 1\}^S\]
LaTeX source
\[
(22) \qquad \gamma(G) = \gamma(G^b) = \{\pm 1\}^S
\]\[(23) \qquad 0 \to \mathbb{Z}^S \xrightarrow{\ 2\ } \mathbb{Z}^S \to \underset{\gamma(G)}{(\mathbb{Z}/2\mathbb{Z})^S} \to 0\]
LaTeX source
\[
(23) \qquad 0 \to \mathbb{Z}^S \xrightarrow{\ 2\ } \mathbb{Z}^S \to \underset{\gamma(G)}{(\mathbb{Z}/2\mathbb{Z})^S} \to 0
\]\[\mathbb{Z}^S \longrightarrow \mathbb{Z}^{\vec{\hat A}}\]
LaTeX source
\[
\mathbb{Z}^S \longrightarrow \mathbb{Z}^{\vec{\hat A}}
\]\[\vec{\hat A} \longrightarrow S\]
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\[
\vec{\hat A} \longrightarrow S
\]\[(24) \qquad 0 \to \mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathcal{E}(G^b) \longrightarrow \gamma(G) \to 0\]
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\[
(24) \qquad 0 \to \mathbb{Z}^{\vec{\hat A}} \longrightarrow \mathcal{E}(G^b) \longrightarrow \gamma(G) \to 0
\]\[(25) \qquad S_0\mathcal{T}_{G^b} = \mathcal{E}(G^b)(\underbrace{\varepsilon(G^b)}_{\varepsilon(G)})\]
LaTeX source
\[
(25) \qquad S_0\mathcal{T}_{G^b} = \mathcal{E}(G^b)(\underbrace{\varepsilon(G^b)}_{\varepsilon(G)})
\]\[(26) \qquad 0 \to \mathbb{Z}^{\hat A} \longrightarrow \mathcal{E}(G) \longrightarrow \gamma(G) \to 0\]
LaTeX source
\[
(26) \qquad 0 \to \mathbb{Z}^{\hat A} \longrightarrow \mathcal{E}(G) \longrightarrow \gamma(G) \to 0
\]\[(27) \qquad S_0\mathcal{T}_G \simeq \mathcal{E}(G)(\varepsilon(G))\]
LaTeX source
\[
(27) \qquad S_0\mathcal{T}_G \simeq \mathcal{E}(G)(\varepsilon(G))
\]\[(28) \qquad
\begin{cases}
\mathrm{Ob}\, S_0\mathcal{T}_G = \varepsilon(G) \\
\mathrm{Hom}_{S_0\mathcal{T}_G}(x,y) = \underbrace{\mathcal{E}(G)_{y-x}}_{\text{fibre de $\mathcal{E}(G)$ au-dessus de $y - x \in \gamma(G)$}}
\quad \text{pour } x, y \in \varepsilon(G)
\end{cases}\]
LaTeX source
\[
(28) \qquad
\begin{cases}
\mathrm{Ob}\, S_0\mathcal{T}_G = \varepsilon(G) \\
\mathrm{Hom}_{S_0\mathcal{T}_G}(x,y) = \underbrace{\mathcal{E}(G)_{y-x}}_{\text{fibre de $\mathcal{E}(G)$ au-dessus de $y - x \in \gamma(G)$}}
\quad \text{pour } x, y \in \varepsilon(G)
\end{cases}
\]\[A(Q) = \prod_{\Delta \in \mathrm{diag}\,Q} \Delta\]
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\[
A(Q) = \prod_{\Delta \in \mathrm{diag}\,Q} \Delta
\]\[\mathrm{codiag}(Q) \simeq \bigwedge_{\Delta \in \mathrm{diag}\,Q} \Delta .\]
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\[
\mathrm{codiag}(Q) \simeq \bigwedge_{\Delta \in \mathrm{diag}\,Q} \Delta .
\]\[\begin{aligned}
S &\simeq \mathrm{diag}\,Q && \text{soit $\Delta_s$ la diagonale associée à $s \in S$} \\
I = A^{\mathrm{lib}} &= A(Q) && \text{ens.\ des sommets de $Q$} \\
\vec{\hat A} &\simeq A \amalg \tilde S && \text{en associant à $s \in S$ l'arête orientée $\tilde s$ d'origine $s$}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
S &\simeq \mathrm{diag}\,Q && \text{soit $\Delta_s$ la diagonale associée à $s \in S$} \\
I = A^{\mathrm{lib}} &= A(Q) && \text{ens.\ des sommets de $Q$} \\
\vec{\hat A} &\simeq A \amalg \tilde S && \text{en associant à $s \in S$ l'arête orientée $\tilde s$ d'origine $s$}
\end{aligned}
\]\[\vec{\hat A}_s = \Delta_s \amalg \{\tilde s\}
\qquad \text{donc} \qquad
\varepsilon_s = \omega(\vec{\hat A}_s) \simeq \Delta_s\]
LaTeX source
\[
\vec{\hat A}_s = \Delta_s \amalg \{\tilde s\}
\qquad \text{donc} \qquad
\varepsilon_s = \omega(\vec{\hat A}_s) \simeq \Delta_s
\]\[\varepsilon(G) \simeq \prod_{s\in S} \Delta_s \simeq A(Q)
\qquad \text{en tant que torseurs sous}\]
LaTeX source
\[
\varepsilon(G) \simeq \prod_{s\in S} \Delta_s \simeq A(Q)
\qquad \text{en tant que torseurs sous}
\]\[\gamma_G = \prod_{s\in S} \{\pm1\} = \{\pm1\}^S .\]
LaTeX source
\[
\gamma_G = \prod_{s\in S} \{\pm1\} = \{\pm1\}^S .
\]\[0 \longrightarrow \mathbb{Z}^S \xrightarrow{\ 2\ } \mathbb{Z}^S \longrightarrow \overbrace{\{\pm1\}^S}^{\gamma(G)} \longrightarrow 0\]
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\[
0 \longrightarrow \mathbb{Z}^S \xrightarrow{\ 2\ } \mathbb{Z}^S \longrightarrow \overbrace{\{\pm1\}^S}^{\gamma(G)} \longrightarrow 0
\]\[\mathbb{Z}^S \to \mathbb{Z}^I \quad \text{associé par contravariance à } I \xrightarrow[\deg 2]{} S\]
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\[
\mathbb{Z}^S \to \mathbb{Z}^I \quad \text{associé par contravariance à } I \xrightarrow[\deg 2]{} S
\]\[\mathbb{Z}^S \xrightarrow[\text{homm.\ somme}]{} \mathbb{Z} .\]
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\[
\mathbb{Z}^S \xrightarrow[\text{homm.\ somme}]{} \mathbb{Z} .
\]\[(1) \qquad
\begin{cases}
J = \text{ens.\ des partitions de type $(2,2)$ de $I$} \\
\Sigma = \mathbb{P}^1_J \quad \text{la \struck{sphère} \add{dr.\ proj.} $J$-marquée associée} \\
\omega = \omega(J) \quad \text{l'ens.\ des orientations de $J$.}
\end{cases}\]
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\[
(1) \qquad
\begin{cases}
J = \text{ens.\ des partitions de type $(2,2)$ de $I$} \\
\Sigma = \mathbb{P}^1_J \quad \text{la \struck{sphère} \add{dr.\ proj.} $J$-marquée associée} \\
\omega = \omega(J) \quad \text{l'ens.\ des orientations de $J$.}
\end{cases}
\]\[(2) \qquad \omega \simeq \Sigma^{\mathfrak{S}_J^+} = \Sigma^{\mathfrak{S}_I^+}\]
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\[
(2) \qquad \omega \simeq \Sigma^{\mathfrak{S}_J^+} = \Sigma^{\mathfrak{S}_I^+}
\]\[(3) \qquad \Sigma^* = \Sigma_I^* = \Sigma \setminus J\]
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\[ (3) \qquad \Sigma^* = \Sigma_I^* = \Sigma \setminus J \]
\[(4) \qquad
\begin{cases}
M_{0,I} \simeq \Sigma_I^* \\
\hat M_{0,I} \simeq \Sigma_I
\end{cases}\]
LaTeX source
\[
(4) \qquad
\begin{cases}
M_{0,I} \simeq \Sigma_I^* \\
\hat M_{0,I} \simeq \Sigma_I
\end{cases}
\]\[\Sigma_J = \mathbb{P}(V(J)) \simeq \check{\mathbb{P}}(V(J))\]
LaTeX source
\[
\Sigma_J = \mathbb{P}(V(J)) \simeq \check{\mathbb{P}}(V(J))
\]\[0 \to V(J) \longrightarrow \mathcal{O}^J \xrightarrow[\text{somme}]{} \mathcal{O} \to 0\]
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\[
0 \to V(J) \longrightarrow \mathcal{O}^J \xrightarrow[\text{somme}]{} \mathcal{O} \to 0
\]\[V(J) \xrightarrow{\ \mathrm{pr}_\alpha\ } \mathcal{O}\]
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\[
V(J) \xrightarrow{\ \mathrm{pr}_\alpha\ } \mathcal{O}
\]\[-2\alpha + \beta + \gamma = -2\alpha + \sum_{\beta \in J \setminus \{\alpha\}} \beta\]
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\[
-2\alpha + \beta + \gamma = -2\alpha + \sum_{\beta \in J \setminus \{\alpha\}} \beta
\]\[0 \to \Delta_{\alpha'} \longrightarrow V(J) \longrightarrow \mathcal{O}_{\alpha'}(1) \to 0\]
LaTeX source
\[
0 \to \Delta_{\alpha'} \longrightarrow V(J) \longrightarrow \mathcal{O}_{\alpha'}(1) \to 0
\]\[\Delta_{\alpha'} \xrightarrow[\substack{\mathrm{pr}_\beta \\ (\beta \neq \alpha)}]{\ \sim\ } \mathcal{O} .\]
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\[
\Delta_{\alpha'} \xrightarrow[\substack{\mathrm{pr}_\beta \\ (\beta \neq \alpha)}]{\ \sim\ } \mathcal{O} .
\]\[\det V(J) \simeq \mathcal{O}_{\alpha'}(1) \simeq \mathcal{O}(1) \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)\]
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\[
\det V(J) \simeq \mathcal{O}_{\alpha'}(1) \simeq \mathcal{O}(1) \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)
\]\[\hat T_i \simeq \mathcal{O}(1)^{\otimes} \otimes_{\mathbb{Z}} L(\omega) ,\]
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\[
\hat T_i \simeq \mathcal{O}(1)^{\otimes} \otimes_{\mathbb{Z}} L(\omega) ,
\]\[\begin{cases}
L(\omega) = \mathbb{Z} \\
L(\omega) = \mathbb{Z}(\omega)
\end{cases}\]
LaTeX source
\[
\begin{cases}
L(\omega) = \mathbb{Z} \\
L(\omega) = \mathbb{Z}(\omega)
\end{cases}
\]\[\underset{\substack{\text{fibré tangent} \\ \text{en long de la} \\ \text{section $i$ de la} \\ \text{droite proj.\ relative} \\ \text{$X$ $I$-marquée}}}{T_i}
\simeq \underbrace{\underbrace{\mathcal{O}_{\alpha'}(1)}_{\mathcal{O} \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)} \otimes_{\mathbb{Z}} L(\omega)}_{\mathcal{O} \otimes_{\mathbb{Z}} (\mathbb{Z}(\omega) \otimes_{\mathbb{Z}} L(\omega))}\]
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\[
\underset{\substack{\text{fibré tangent} \\ \text{en long de la} \\ \text{section $i$ de la} \\ \text{droite proj.\ relative} \\ \text{$X$ $I$-marquée}}}{T_i}
\simeq \underbrace{\underbrace{\mathcal{O}_{\alpha'}(1)}_{\mathcal{O} \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)} \otimes_{\mathbb{Z}} L(\omega)}_{\mathcal{O} \otimes_{\mathbb{Z}} (\mathbb{Z}(\omega) \otimes_{\mathbb{Z}} L(\omega))}
\]\[\omega(Q) \simeq \omega\]
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\[ \omega(Q) \simeq \omega \]
\[T_i \simeq \mathcal{O} \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)\]
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\[
T_i \simeq \mathcal{O} \otimes_{\mathbb{Z}} \mathbb{Z}(\omega)
\]\[\boxed{\hat T_i \simeq \mathcal{O}(1)} \simeq \check F(\omega)
\qquad \text{où } 0 \to F \to V(J) \to \mathcal{O}_\Sigma(1) \to 0 .\]
LaTeX source
\[
\boxed{\hat T_i \simeq \mathcal{O}(1)} \simeq \check F(\omega)
\qquad \text{où } 0 \to F \to V(J) \to \mathcal{O}_\Sigma(1) \to 0 .
\]\[S\hat P_I \overset{\mathrm{déf}}{=} \prod_{/\hat M_{0,I}} \hat P_i
\simeq \bigl(\check V^*(\mathcal{O}(1))^{I}\bigr)_{/\underset{\Sigma_I}{\hat M_{0,I}}}
\simeq V^*(\mathcal{O}(1)) \wedge^{\mathbb{G}_m} \mathbb{G}_m^I\]
LaTeX source
\[
S\hat P_I \overset{\mathrm{déf}}{=} \prod_{/\hat M_{0,I}} \hat P_i
\simeq \bigl(\check V^*(\mathcal{O}(1))^{I}\bigr)_{/\underset{\Sigma_I}{\hat M_{0,I}}}
\simeq V^*(\mathcal{O}(1)) \wedge^{\mathbb{G}_m} \mathbb{G}_m^I
\]\[SP_I \simeq \bigl(V^*(\mathcal{O}(1)) \mid \underset{\Sigma_I^*}{M_{0,I}}\bigr) \wedge^{\mathbb{G}_m} \mathbb{G}_m^I\]
LaTeX source
\[
SP_I \simeq \bigl(V^*(\mathcal{O}(1)) \mid \underset{\Sigma_I^*}{M_{0,I}}\bigr) \wedge^{\mathbb{G}_m} \mathbb{G}_m^I
\]\[S\mathcal{T}_{0,I} \simeq \Pi_1\bigl(V^*(\mathcal{O}(1)) \mid \Sigma_I^*\bigr) \wedge^{\mathbb{Z}} \mathbb{Z}^I\]
LaTeX source
\[
S\mathcal{T}_{0,I} \simeq \Pi_1\bigl(V^*(\mathcal{O}(1)) \mid \Sigma_I^*\bigr) \wedge^{\mathbb{Z}} \mathbb{Z}^I
\]\[\begin{aligned}
& s \in S, \qquad a \in A, \qquad r = (s, a) \\
& r_1 = (s, a_1) = \tau_1 r \\
& r_2 = \omega^{-1}(r_1) = \tau_0 \tau_1 r = \tau_0 \tau_1 r \\
& r' = \tau_0 r \\
& r'' = \tau_1 r' = \tau_1 \tau_0 r = \omega(r)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
& s \in S, \qquad a \in A, \qquad r = (s, a) \\
& r_1 = (s, a_1) = \tau_1 r \\
& r_2 = \omega^{-1}(r_1) = \tau_0 \tau_1 r = \tau_0 \tau_1 r \\
& r' = \tau_0 r \\
& r'' = \tau_1 r' = \tau_1 \tau_0 r = \omega(r)
\end{aligned}
\]\[\widetilde{R}_i = (e_j - e_k)\wedge\omega_{jk}, \qquad
\widetilde{Q}^\omega_i = \widetilde{R}_{\omega^2 i} - \widetilde{R}_{\omega i},\]
LaTeX source
\[
\widetilde{R}_i = (e_j - e_k)\wedge\omega_{jk}, \qquad
\widetilde{Q}^\omega_i = \widetilde{R}_{\omega^2 i} - \widetilde{R}_{\omega i},
\]\[i_\omega \widetilde{Q}^\omega_i = \widetilde{R}_i, \qquad
i_\omega \widetilde{R}_i = -\widetilde{Q}^\omega_i = \widetilde{Q}^{\omega'}_i ,\]
LaTeX source
\[
i_\omega \widetilde{Q}^\omega_i = \widetilde{R}_i, \qquad
i_\omega \widetilde{R}_i = -\widetilde{Q}^\omega_i = \widetilde{Q}^{\omega'}_i ,
\]\[\widetilde{Q}^\omega_i = (e_j + e_k - 2e_i)\wedge\omega .\]
LaTeX source
\[
\widetilde{Q}^\omega_i = (e_j + e_k - 2e_i)\wedge\omega .
\]\[\begin{array}{lll}
a_r \leftrightarrow \tilde a_r : \widetilde{R}_r \to \widetilde{R}_{r'} & & r \in R\\
b_r \leftrightarrow \tilde b_r : \widetilde{Q}_a \to \widetilde{R}_r & & r\in R\\
c_a \leftrightarrow \tilde c^\omega_a : \widetilde{P}^\omega_a \to \widetilde{Q}_a \ \text{ou}\ \tilde c_r & & (a,\omega)\in A\times\underline{\omega}\simeq R
\end{array}\]
LaTeX source
\[
\begin{array}{lll}
a_r \leftrightarrow \tilde a_r : \widetilde{R}_r \to \widetilde{R}_{r'} & & r \in R\\
b_r \leftrightarrow \tilde b_r : \widetilde{Q}_a \to \widetilde{R}_r & & r\in R\\
c_a \leftrightarrow \tilde c^\omega_a : \widetilde{P}^\omega_a \to \widetilde{Q}_a \ \text{ou}\ \tilde c_r & & (a,\omega)\in A\times\underline{\omega}\simeq R
\end{array}
\]\[\begin{array}{lll}
1 \leftrightarrow \lambda_{\widetilde{P}^\omega_a}\ \text{lacet en}\ \widetilde{P}^\omega_a & & (a,\omega)\in A\times\underline{\omega}\simeq R\\
1 \leftrightarrow \lambda_{\widetilde{Q}_a}\ \text{lacet en}\ \widetilde{Q}_a & & a \in A\\
1 \leftrightarrow \lambda_{\widetilde{R}_r}\ \text{lacet en}\ \widetilde{R}_r & & r\in R\\
\mu_{\widetilde{Q}_a} : \widetilde{Q}_a \to \widetilde{Q}_a^\vee & & a\in A\\
\mu_{\widetilde{R}_r} : \widetilde{R}_r \to \widetilde{R}_r^\vee & & r\in R\\
1 \leftrightarrow \tilde d^\omega_r : \widetilde{P}^\omega_{a_1} \to \widetilde{P}^\omega_{\bullet}\ \text{ou}\ \tilde d^\omega_s & & r = (s,\omega)\in S\times\underline{\omega}\simeq R
\end{array}\]
LaTeX source
\[
\begin{array}{lll}
1 \leftrightarrow \lambda_{\widetilde{P}^\omega_a}\ \text{lacet en}\ \widetilde{P}^\omega_a & & (a,\omega)\in A\times\underline{\omega}\simeq R\\
1 \leftrightarrow \lambda_{\widetilde{Q}_a}\ \text{lacet en}\ \widetilde{Q}_a & & a \in A\\
1 \leftrightarrow \lambda_{\widetilde{R}_r}\ \text{lacet en}\ \widetilde{R}_r & & r\in R\\
\mu_{\widetilde{Q}_a} : \widetilde{Q}_a \to \widetilde{Q}_a^\vee & & a\in A\\
\mu_{\widetilde{R}_r} : \widetilde{R}_r \to \widetilde{R}_r^\vee & & r\in R\\
1 \leftrightarrow \tilde d^\omega_r : \widetilde{P}^\omega_{a_1} \to \widetilde{P}^\omega_{\bullet}\ \text{ou}\ \tilde d^\omega_s & & r = (s,\omega)\in S\times\underline{\omega}\simeq R
\end{array}
\]\[(1)\quad \tilde a_r \tilde b_r \tilde c_r \tilde d_r = \tilde b_{r_1} \tilde c_{r_2}
\qquad r\in R,\ r' = \sigma_0 r,\ r_1 = \sigma_1\sigma_0(r') = \omega^{-1} r'\]
LaTeX source
\[
(1)\quad \tilde a_r \tilde b_r \tilde c_r \tilde d_r = \tilde b_{r_1} \tilde c_{r_2}
\qquad r\in R,\ r' = \sigma_0 r,\ r_1 = \sigma_1\sigma_0(r') = \omega^{-1} r'
\]\[(2)\quad \tilde d_{\omega^m r}\cdots \tilde d_{\omega r}\tilde d_r = \lambda_{\widetilde{P}^\omega_{a_1}}
\qquad r\in R,\ r = (s,a,\omega),\ a' = \omega^{-1}(a)\]
LaTeX source
\[
(2)\quad \tilde d_{\omega^m r}\cdots \tilde d_{\omega r}\tilde d_r = \lambda_{\widetilde{P}^\omega_{a_1}}
\qquad r\in R,\ r = (s,a,\omega),\ a' = \omega^{-1}(a)
\]\[(6)\quad \mu_{\widetilde{Q}_a^\vee}\,\mu_{\widetilde{Q}_a} = \lambda_{\widetilde{Q}_a}
\qquad (7)\quad \mu_{\widetilde{R}_r^\vee}\,\mu_{\widetilde{R}_r} = \lambda_{\widetilde{R}_r}
\qquad r\in R\]
LaTeX source
\[
(6)\quad \mu_{\widetilde{Q}_a^\vee}\,\mu_{\widetilde{Q}_a} = \lambda_{\widetilde{Q}_a}
\qquad (7)\quad \mu_{\widetilde{R}_r^\vee}\,\mu_{\widetilde{R}_r} = \lambda_{\widetilde{R}_r}
\qquad r\in R
\]\[\begin{array}{lll}
(8) & \tilde a_r(\lambda_{\widetilde{R}_r}) = \lambda_{\widetilde{R}_{r'}} & r\in R,\ r' = \sigma_0(r)\\
(9) & \tilde b_r(\lambda_{\widetilde{Q}_a}) = \lambda_{\widetilde{R}_r} & r\in R,\ r = (a,s,\omega)\\
(10) & \tilde c_r(\lambda_{\widetilde{P}^\omega_a}) = \lambda_{\widetilde{Q}_a} & (a,\omega)\in A\times\underline{\omega}\simeq R
\end{array}\]
LaTeX source
\[
\begin{array}{lll}
(8) & \tilde a_r(\lambda_{\widetilde{R}_r}) = \lambda_{\widetilde{R}_{r'}} & r\in R,\ r' = \sigma_0(r)\\
(9) & \tilde b_r(\lambda_{\widetilde{Q}_a}) = \lambda_{\widetilde{R}_r} & r\in R,\ r = (a,s,\omega)\\
(10) & \tilde c_r(\lambda_{\widetilde{P}^\omega_a}) = \lambda_{\widetilde{Q}_a} & (a,\omega)\in A\times\underline{\omega}\simeq R
\end{array}
\]\[(3)\quad \mu_{\widetilde{R}_r}\circ\tilde b_r = \tilde b_{r^\vee}\circ\mu_{\widetilde{Q}^\omega_a}\]
LaTeX source
\[
(3)\quad \mu_{\widetilde{R}_r}\circ\tilde b_r = \tilde b_{r^\vee}\circ\mu_{\widetilde{Q}^\omega_a}
\]\[(4)\quad \mu_{\widetilde{R}_{r'}}\circ\tilde a_r = \tilde a_{r^\vee}\,\mu_{\widetilde{R}_r}\]
LaTeX source
\[
(4)\quad \mu_{\widetilde{R}_{r'}}\circ\tilde a_r = \tilde a_{r^\vee}\,\mu_{\widetilde{R}_r}
\]\[(2)\quad \mu_{\widetilde{Q}_a}\,\tilde c_r = \tilde c_{r^\vee}\,\tilde d_{\omega^n r^*}\cdots\tilde d_{r^*}
\ \Longrightarrow\ \ill{}\]
LaTeX source
\[
(2)\quad \mu_{\widetilde{Q}_a}\,\tilde c_r = \tilde c_{r^\vee}\,\tilde d_{\omega^n r^*}\cdots\tilde d_{r^*}
\ \Longrightarrow\ \ill{}
\]\[\overset{\text{via (1)}}{\Longrightarrow}\quad
\mu_{\widetilde{Q}_0} = \widetilde{A}_{\omega^n r^*}\cdots\widetilde{A}_{\omega r^*}\widetilde{A}_{r^*},
\qquad \widetilde{A}_r = \tilde b_r^{-1}\tilde a_r^{-1}\tilde b_{r_1}\]
LaTeX source
\[
\overset{\text{via (1)}}{\Longrightarrow}\quad
\mu_{\widetilde{Q}_0} = \widetilde{A}_{\omega^n r^*}\cdots\widetilde{A}_{\omega r^*}\widetilde{A}_{r^*},
\qquad \widetilde{A}_r = \tilde b_r^{-1}\tilde a_r^{-1}\tilde b_{r_1}
\]\[\mu_{\widetilde{R}_r} = \widetilde{B}_{\omega^n r}\cdots\widetilde{B}_{\omega r}\widetilde{B}_r
\quad\text{où}\quad
\widetilde{B}_r \overset{\text{déf}}{=} \tilde a_{r''}^{-1}\,\tilde b_{r'}\,\tilde b_r^{-1}\]
LaTeX source
\[
\mu_{\widetilde{R}_r} = \widetilde{B}_{\omega^n r}\cdots\widetilde{B}_{\omega r}\widetilde{B}_r
\quad\text{où}\quad
\widetilde{B}_r \overset{\text{déf}}{=} \tilde a_{r''}^{-1}\,\tilde b_{r'}\,\tilde b_r^{-1}
\]\[(2')\quad \widetilde{A}_{\omega^m r^{**}}\cdots\widetilde{A}_{r^{**}} = \widetilde{A}_{\omega^m r_1^*}\cdots\widetilde{A}_{r_1^*}
\qquad \forall a\in A\]
LaTeX source
\[
(2')\quad \widetilde{A}_{\omega^m r^{**}}\cdots\widetilde{A}_{r^{**}} = \widetilde{A}_{\omega^m r_1^*}\cdots\widetilde{A}_{r_1^*}
\qquad \forall a\in A
\]\[\begin{array}{ll}
(1) & \tilde a_r\tilde b_r\tilde c_r\tilde d_r = \tilde b_{r_1}\tilde c_{r_2}\\[2pt]
& \quad r\in R,\ r_1 = \sigma_0(r),\ r_2 = \sigma_1\sigma_0(r) = \omega^{-1}(r)\\[4pt]
(2') & \widetilde{A}_{\omega^m r''}\widetilde{A}_{\omega^{m-1} r''}\cdots\widetilde{A}_{r''} = \widetilde{A}_{\omega^m r_1}\cdots\widetilde{A}_{r_1}\\[2pt]
& \quad r\in R,\ r_1 = \sigma_0 r,\ r'' = \sigma_0\sigma_1(r) = \omega(r)
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
(1) & \tilde a_r\tilde b_r\tilde c_r\tilde d_r = \tilde b_{r_1}\tilde c_{r_2}\\[2pt]
& \quad r\in R,\ r_1 = \sigma_0(r),\ r_2 = \sigma_1\sigma_0(r) = \omega^{-1}(r)\\[4pt]
(2') & \widetilde{A}_{\omega^m r''}\widetilde{A}_{\omega^{m-1} r''}\cdots\widetilde{A}_{r''} = \widetilde{A}_{\omega^m r_1}\cdots\widetilde{A}_{r_1}\\[2pt]
& \quad r\in R,\ r_1 = \sigma_0 r,\ r'' = \sigma_0\sigma_1(r) = \omega(r)
\end{array}
\]\[\begin{aligned}
\mu_{\widetilde{Q}_0} &\overset{\text{déf}}{=} \widetilde{A}_{\omega^m r''}\cdots\widetilde{A}_{r''} = \cdots\\
\mu_{\widetilde{R}_r} &\overset{\text{déf}}{=} \widetilde{B}_{\omega^m r}\cdots\widetilde{B}_r
\qquad\text{où}\quad \widetilde{B}_r \overset{\text{déf}}{=} \tilde a_{r''}^{-1}\tilde b_{r'}\tilde b_r^{-1}\\
\mu_{\widetilde{P}^\omega_a} &\overset{\text{déf}}{=} \tilde d_{\omega^m r''}\cdots\tilde d_{\omega r''}\tilde d_{r''}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\mu_{\widetilde{Q}_0} &\overset{\text{déf}}{=} \widetilde{A}_{\omega^m r''}\cdots\widetilde{A}_{r''} = \cdots\\
\mu_{\widetilde{R}_r} &\overset{\text{déf}}{=} \widetilde{B}_{\omega^m r}\cdots\widetilde{B}_r
\qquad\text{où}\quad \widetilde{B}_r \overset{\text{déf}}{=} \tilde a_{r''}^{-1}\tilde b_{r'}\tilde b_r^{-1}\\
\mu_{\widetilde{P}^\omega_a} &\overset{\text{déf}}{=} \tilde d_{\omega^m r''}\cdots\tilde d_{\omega r''}\tilde d_{r''}
\end{aligned}
\]\[\begin{aligned}
\lambda_{\widetilde{Q}_0} &\overset{\text{déf}}{=} \mu_{\widetilde{Q}_a^\vee}\,\mu_{\widetilde{Q}_a} = \widetilde{A}_{\omega^m r''}\cdots\widetilde{A}_{r''} = \widetilde{A}_{\omega^m r_1}\cdots\widetilde{A}_{r_1}\\
\lambda_{\widetilde{R}_r} &\overset{\text{déf}}{=} \mu_{\widetilde{R}_r^\vee}\,\mu_{\widetilde{R}_r} = \widetilde{B}_{\omega^m r}\cdots\widetilde{B}_r\\
\lambda_{\widetilde{P}^\omega_a} &\overset{\text{déf}}{=} \mu_{\widetilde{P}^\omega_{a^\vee}}\,\mu_{\widetilde{P}^\omega_a} = \tilde d_{\omega^m r''}\cdots\tilde d_{r''}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\lambda_{\widetilde{Q}_0} &\overset{\text{déf}}{=} \mu_{\widetilde{Q}_a^\vee}\,\mu_{\widetilde{Q}_a} = \widetilde{A}_{\omega^m r''}\cdots\widetilde{A}_{r''} = \widetilde{A}_{\omega^m r_1}\cdots\widetilde{A}_{r_1}\\
\lambda_{\widetilde{R}_r} &\overset{\text{déf}}{=} \mu_{\widetilde{R}_r^\vee}\,\mu_{\widetilde{R}_r} = \widetilde{B}_{\omega^m r}\cdots\widetilde{B}_r\\
\lambda_{\widetilde{P}^\omega_a} &\overset{\text{déf}}{=} \mu_{\widetilde{P}^\omega_{a^\vee}}\,\mu_{\widetilde{P}^\omega_a} = \tilde d_{\omega^m r''}\cdots\tilde d_{r''}
\end{aligned}
\]\[(2'')\quad \tilde c^\omega_a(\lambda_{\widetilde{P}^\omega_a}) = \tilde c^\omega_a(\lambda_{\widetilde{P}^{\omega'}_a})\]
LaTeX source
\[
(2'')\quad \tilde c^\omega_a(\lambda_{\widetilde{P}^\omega_a}) = \tilde c^\omega_a(\lambda_{\widetilde{P}^{\omega'}_a})
\]\[\underbrace{\check\mu\,\mu}_{\lambda} = \underbrace{\check\mu'\,\mu'}_{\lambda'}
\qquad
\check\mu'^{-1}\check\mu = \mu'\mu^{-1} \quad (\mu'^{-1}\mu)^{-1}\]
LaTeX source
\[
\underbrace{\check\mu\,\mu}_{\lambda} = \underbrace{\check\mu'\,\mu'}_{\lambda'}
\qquad
\check\mu'^{-1}\check\mu = \mu'\mu^{-1} \quad (\mu'^{-1}\mu)^{-1}
\]\[\mu' = 1,\ \check\mu' = \check\mu \quad\Longleftarrow\quad \check\mu = \check\mu'\mu'\]
LaTeX source
\[ \mu' = 1,\ \check\mu' = \check\mu \quad\Longleftarrow\quad \check\mu = \check\mu'\mu' \]
\[\overbrace{\widetilde{A}_{\omega^n r''}\widetilde{A}_{\omega^{n-1} r''}\cdots\widetilde{A}_{r''}}^{\mu_{\widetilde{Q}^\omega_a}}
= \overbrace{\widetilde{A}_{\omega^n r_1}\widetilde{A}_{\omega^{n-1} r_1}\cdots\widetilde{A}_{r_1}}^{\mu_{\widetilde{Q}^{\omega'}_{\bullet}}}\]
LaTeX source
\[
\overbrace{\widetilde{A}_{\omega^n r''}\widetilde{A}_{\omega^{n-1} r''}\cdots\widetilde{A}_{r''}}^{\mu_{\widetilde{Q}^\omega_a}}
= \overbrace{\widetilde{A}_{\omega^n r_1}\widetilde{A}_{\omega^{n-1} r_1}\cdots\widetilde{A}_{r_1}}^{\mu_{\widetilde{Q}^{\omega'}_{\bullet}}}
\]\[f_{x,y}(s,t) = (1-s)\,\varphi_\omega\,\ell_{y,x}(t) + s\,\ell_{y,x}(t),\]
LaTeX source
\[
f_{x,y}(s,t) = (1-s)\,\varphi_\omega\,\ell_{y,x}(t) + s\,\ell_{y,x}(t),
\]\[\varphi_\omega(\ell_{y,x}) = \ell_{\varphi_\omega(y),\varphi_\omega(x)}\]
LaTeX source
\[
\varphi_\omega(\ell_{y,x}) = \ell_{\varphi_\omega(y),\varphi_\omega(x)}
\]\[g(s,t) = \text{\struck{$1$}}\ (1-s)\,\varphi_\omega\,\ell(t) + s\,\ell(t).\]
LaTeX source
\[
g(s,t) = \text{\struck{$1$}}\ (1-s)\,\varphi_\omega\,\ell(t) + s\,\ell(t).
\]\[\ell_x(t) = \ell_{-x,x}(t) = e(t)\,x : x\to -x,\]
LaTeX source
\[
\ell_x(t) = \ell_{-x,x}(t) = e(t)\,x : x\to -x,
\]\[\begin{aligned}
h(s,t) &= (1-s)\ \text{\struck{$\cdots$}}\ \ell_{\varphi(x)}(t) + s\,\ell_x(t)\\
&= e(t)\bigl((1-s)\varphi(x) + s x\bigr).
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
h(s,t) &= (1-s)\ \text{\struck{$\cdots$}}\ \ell_{\varphi(x)}(t) + s\,\ell_x(t)\\
&= e(t)\bigl((1-s)\varphi(x) + s x\bigr).
\end{aligned}
\]\[h([0,1]\times[0,1])\cap\Delta_{\mathbb C} = \emptyset.\]
LaTeX source
\[
h([0,1]\times[0,1])\cap\Delta_{\mathbb C} = \emptyset.
\]\[(1)\qquad D_m = \mu_m \cdot \mathbb{Z}/2\mathbb{Z}\]
LaTeX source
\[
(1)\qquad D_m = \mu_m \cdot \mathbb{Z}/2\mathbb{Z}
\]\[(2)\qquad V_0 = \mathbb{C}, \qquad \tilde{J}_0 = \mu_m
\overset{\text{déf}}{=} {}_m\mathbb{C}^{*} \subset V_0 ,\]
LaTeX source
\[
(2)\qquad V_0 = \mathbb{C}, \qquad \tilde{J}_0 = \mu_m
\overset{\text{déf}}{=} {}_m\mathbb{C}^{*} \subset V_0 ,
\]\[z \longmapsto \zeta z \qquad (\zeta \in \mu_m)\]
LaTeX source
\[ z \longmapsto \zeta z \qquad (\zeta \in \mu_m) \]
\[(3)\qquad D = D_{\tilde{J}} = \operatorname{Aut}(V, \tilde{J})
\qquad (\tilde{J} \subset V)\]
LaTeX source
\[
(3)\qquad D = D_{\tilde{J}} = \operatorname{Aut}(V, \tilde{J})
\qquad (\tilde{J} \subset V)
\]\[(4)\qquad \underline{\omega} = \underline{\omega}_{\tilde{J}}
= \Omega(\tilde{J}) \simeq \Omega(V)\]
LaTeX source
\[
(4)\qquad \underline{\omega} = \underline{\omega}_{\tilde{J}}
= \Omega(\tilde{J}) \simeq \Omega(V)
\]\[(5)\qquad 1 \longrightarrow D^{+} \longrightarrow D \longrightarrow
\mathbb{Z}/2\mathbb{Z} \longrightarrow 1 ,\]
LaTeX source
\[
(5)\qquad 1 \longrightarrow D^{+} \longrightarrow D \longrightarrow
\mathbb{Z}/2\mathbb{Z} \longrightarrow 1 ,
\]\[(6)\qquad \underline{\omega} \subset D^{+}\]
LaTeX source
\[
(6)\qquad \underline{\omega} \subset D^{+}
\]\[(7)\qquad a_0 \in D^{+}, \qquad a_0 : x \longmapsto -x ,\]
LaTeX source
\[
(7)\qquad a_0 \in D^{+}, \qquad a_0 : x \longmapsto -x ,
\]\[(8)\qquad r_j(\tilde{R}_j) = \tilde{R}_j ,\]
LaTeX source
\[
(8)\qquad r_j(\tilde{R}_j) = \tilde{R}_j ,
\]\[(9)\qquad r_j = r_{a_0(j)} ,\]
LaTeX source
\[
(9)\qquad r_j = r_{a_0(j)} ,
\]\[\text{\struck{$\Sigma \ldots$}}\]
LaTeX source
\[
\text{\struck{$\Sigma \ldots$}}
\]\[(10)\qquad \Sigma_{\tilde{J}}(\mathbb{C}) = \Sigma_{\tilde{J}}
\overset{\text{déf}}{=} \Sigma = \mathbb{P}(V_{\mathbb{C}})
= V_{\mathbb{C}}^{*}/\mathbb{C}^{*} ,\]
LaTeX source
\[
(10)\qquad \Sigma_{\tilde{J}}(\mathbb{C}) = \Sigma_{\tilde{J}}
\overset{\text{déf}}{=} \Sigma = \mathbb{P}(V_{\mathbb{C}})
= V_{\mathbb{C}}^{*}/\mathbb{C}^{*} ,
\]\[(14)\qquad \pi_0(\Sigma \setminus \Sigma_{\mathbb{R}}) \simeq
\underline{\omega} = \Omega(\Sigma_{\mathbb{R}})\]
LaTeX source
\[
(14)\qquad \pi_0(\Sigma \setminus \Sigma_{\mathbb{R}}) \simeq
\underline{\omega} = \Omega(\Sigma_{\mathbb{R}})
\]\[(15)\qquad
\begin{cases}
P_\omega \quad (\omega \in \underline{\omega}) \\
P_\omega = \Sigma^{D^{+}} \cap \Sigma_\omega
\end{cases}\]
LaTeX source
\[
(15)\qquad
\begin{cases}
P_\omega \quad (\omega \in \underline{\omega}) \\
P_\omega = \Sigma^{D^{+}} \cap \Sigma_\omega
\end{cases}
\]\[(16)\qquad V_{\mathbb{C}} = \bigoplus_{\omega \in \underline{\omega}}
V_{\mathbb{C}}^{\omega}\]
LaTeX source
\[
(16)\qquad V_{\mathbb{C}} = \bigoplus_{\omega \in \underline{\omega}}
V_{\mathbb{C}}^{\omega}
\]\[(17)\qquad
\begin{cases}
u_\omega \in \text{\struck{\ill{}}}\; O^{+}(V) \\
u_\omega^{2} = -1
\end{cases}\]
LaTeX source
\[
(17)\qquad
\begin{cases}
u_\omega \in \text{\struck{\ill{}}}\; O^{+}(V) \\
u_\omega^{2} = -1
\end{cases}
\]\[i_\omega\, x = u_\omega x .\]
LaTeX source
\[ i_\omega\, x = u_\omega x . \]
\[(17')\qquad V_{\mathbb{C}} \xrightarrow{\;p_\omega\;} V_{(\omega)}
\qquad (\mathbb{C}\text{-lin.})\]
LaTeX source
\[
(17')\qquad V_{\mathbb{C}} \xrightarrow{\;p_\omega\;} V_{(\omega)}
\qquad (\mathbb{C}\text{-lin.})
\]\[(18)\qquad p = (p_\omega)_{\omega \in \underline{\omega}} : V_{\mathbb{C}}
\xrightarrow{\;\sim\;} \prod_{\omega \in \underline{\omega}}
\underbrace{V_{(\omega)}}_{\simeq V_{\mathbb{C}}^{\omega}} .\]
LaTeX source
\[
(18)\qquad p = (p_\omega)_{\omega \in \underline{\omega}} : V_{\mathbb{C}}
\xrightarrow{\;\sim\;} \prod_{\omega \in \underline{\omega}}
\underbrace{V_{(\omega)}}_{\simeq V_{\mathbb{C}}^{\omega}} .
\]\[(19)\qquad
\begin{aligned}
V_{\mathbb{C}}^{\omega}
&= \operatorname{Ker}\bigl(V_{\mathbb{C}} \xrightarrow{\;p_{\omega'}\;}
V_{(\omega')}\bigr) \\
&= \{\, x \in V_{\mathbb{C}} \mid u_{\omega,\mathbb{C}}(x) = i.x \,\}
\qquad (\omega \in \underline{\omega})
\end{aligned}\]
LaTeX source
\[
(19)\qquad
\begin{aligned}
V_{\mathbb{C}}^{\omega}
&= \operatorname{Ker}\bigl(V_{\mathbb{C}} \xrightarrow{\;p_{\omega'}\;}
V_{(\omega')}\bigr) \\
&= \{\, x \in V_{\mathbb{C}} \mid u_{\omega,\mathbb{C}}(x) = i.x \,\}
\qquad (\omega \in \underline{\omega})
\end{aligned}
\]\[(18)\qquad
\begin{cases}
V(\underline{\mathcal{O}}(1))_x \simeq V/F
\qquad \bigl(\mathcal{O}_F(1) \overset{\text{déf}}{=} \ldots\bigr) \\
F \otimes \mathcal{O}_F(1) \simeq \det(V) \simeq
\mathbb{R}(\underline{\omega})
\;\bigl(\overset{\text{déf}}{=} \mathbb{R} \wedge_{\{\pm1\}}
\underline{\omega}\bigr) \\
\mathcal{O}_F(1) \simeq \check{F}(\omega)
\;\bigl(\overset{\text{déf}}{=} \check{F} \wedge_{\{\pm1\}}
\underline{\omega}\bigr) \simeq F^{\perp}
\end{cases}\]
LaTeX source
\[
(18)\qquad
\begin{cases}
V(\underline{\mathcal{O}}(1))_x \simeq V/F
\qquad \bigl(\mathcal{O}_F(1) \overset{\text{déf}}{=} \ldots\bigr) \\
F \otimes \mathcal{O}_F(1) \simeq \det(V) \simeq
\mathbb{R}(\underline{\omega})
\;\bigl(\overset{\text{déf}}{=} \mathbb{R} \wedge_{\{\pm1\}}
\underline{\omega}\bigr) \\
\mathcal{O}_F(1) \simeq \check{F}(\omega)
\;\bigl(\overset{\text{déf}}{=} \check{F} \wedge_{\{\pm1\}}
\underline{\omega}\bigr) \simeq F^{\perp}
\end{cases}
\]\[(19)\qquad (P\Sigma)_x = (P\Sigma)_F \simeq (V/F)^{*} \simeq
(\check{F})^{*} \wedge_{\{\pm1\}} \underline{\omega} \simeq F^{\perp}\]
LaTeX source
\[
(19)\qquad (P\Sigma)_x = (P\Sigma)_F \simeq (V/F)^{*} \simeq
(\check{F})^{*} \wedge_{\{\pm1\}} \underline{\omega} \simeq F^{\perp}
\]\[(20)\qquad (P\Sigma)_x \simeq F^{*} \wedge_{\{\pm1\}} \underline{\omega}
= (F \wedge_{\{\pm1\}} \underline{\omega})^{*} \simeq (F^{\perp})^{*}\]
LaTeX source
\[
(20)\qquad (P\Sigma)_x \simeq F^{*} \wedge_{\{\pm1\}} \underline{\omega}
= (F \wedge_{\{\pm1\}} \underline{\omega})^{*} \simeq (F^{\perp})^{*}
\]\[(21)\qquad
\begin{cases}
P\Sigma \simeq (V(\omega))^{*} \\
P\Sigma \simeq V^{*}
\end{cases}\]
LaTeX source
\[
(21)\qquad
\begin{cases}
P\Sigma \simeq (V(\omega))^{*} \\
P\Sigma \simeq V^{*}
\end{cases}
\]\[V \simeq \check{V}(\omega)\]
LaTeX source
\[
V \simeq \check{V}(\omega)
\]\[(21)\qquad V \xrightarrow{\;\sim\;} V(\omega)\]
LaTeX source
\[
(21)\qquad V \xrightarrow{\;\sim\;} V(\omega)
\]\[\text{\struck{$\Sigma \simeq P(V) \xrightarrow{\text{id}}
\Sigma \simeq P(V(\omega))$}}\]
LaTeX source
\[
\text{\struck{$\Sigma \simeq P(V) \xrightarrow{\text{id}}
\Sigma \simeq P(V(\omega))$}}
\]\[(22)\qquad \Sigma = P(V) \xrightarrow{\;\sigma\;} P(V(\omega))
\qquad (\simeq \Sigma = P(V))\]
LaTeX source
\[
(22)\qquad \Sigma = P(V) \xrightarrow{\;\sigma\;} P(V(\omega))
\qquad (\simeq \Sigma = P(V))
\]\[V \xrightarrow{\;\sim\;} \ldots \Sigma \simeq V^{*}\]
LaTeX source
\[
V \xrightarrow{\;\sim\;} \ldots \Sigma \simeq V^{*}
\]\[P\Sigma \simeq V(\underline{\omega})^{*} ,\]
LaTeX source
\[
P\Sigma \simeq V(\underline{\omega})^{*} ,
\]\[\tau\sigma = \underline{a}\]
LaTeX source
\[
\tau\sigma = \underline{a}
\]\[(26)\qquad \Sigma^{*} = \Sigma \setminus \{\, R_i \mid i \in J \,\}\]
LaTeX source
\[
(26)\qquad \Sigma^{*} = \Sigma \setminus \{\, R_i \mid i \in J \,\}
\]\[(27)\qquad P\Sigma^{*} = P\Sigma \,|\, \Sigma^{*}
\overset{(23)}{\simeq} V^{**} = V^{*} - \bigcup_{i \in J} \Delta_i\]
LaTeX source
\[
(27)\qquad P\Sigma^{*} = P\Sigma \,|\, \Sigma^{*}
\overset{(23)}{\simeq} V^{**} = V^{*} - \bigcup_{i \in J} \Delta_i
\]\[(28)\qquad \Pi_1 P\Sigma^{*} \simeq \Pi_1 V^{**} , \qquad \Pi_1 \Sigma^{*}\]
LaTeX source
\[
(28)\qquad \Pi_1 P\Sigma^{*} \simeq \Pi_1 V^{**} , \qquad \Pi_1 \Sigma^{*}
\]\[(29)\qquad \Pi_1 P\Sigma^{*} = \Pi_1 V^{**} \longrightarrow \Pi_1 \Sigma^{*} .\]
LaTeX source
\[
(29)\qquad \Pi_1 P\Sigma^{*} = \Pi_1 V^{**} \longrightarrow \Pi_1 \Sigma^{*} .
\]\[\text{\struck{\ill{}}}\; D \times \mathbb{C}^{*} \times \{1, \tau\}\]
LaTeX source
\[
\text{\struck{\ill{}}}\; D \times \mathbb{C}^{*} \times \{1, \tau\}
\]\[D \times \mu_m \times \{1, \tau\}\]
LaTeX source
\[
D \times \mu_m \times \{1, \tau\}
\]\[(31)\qquad D^{+} \times \mu_m \simeq \mu_m(\underline{\omega}) \times \mu_m\]
LaTeX source
\[
(31)\qquad D^{+} \times \mu_m \simeq \mu_m(\underline{\omega}) \times \mu_m
\]\[(32)\qquad (\lambda_\omega)_{\omega \in \underline{\omega}} \cdot
(x_\omega)_{\omega \in \underline{\omega}} = (\lambda_\omega x_\omega)\]
LaTeX source
\[
(32)\qquad (\lambda_\omega)_{\omega \in \underline{\omega}} \cdot
(x_\omega)_{\omega \in \underline{\omega}} = (\lambda_\omega x_\omega)
\]\[(33)\qquad
\begin{cases}
\mu_m \longrightarrow \mu_m^{\underline{\omega}}
\qquad \text{hom.\ diagonal} \\
\mu_m(\underline{\omega}) \simeq D^{+} \longrightarrow
\mu_m^{\underline{\omega}} \\
\qquad \lambda \longmapsto (\lambda \wedge \omega)_{\omega \in
\underline{\omega}}
\end{cases}\]
LaTeX source
\[
(33)\qquad
\begin{cases}
\mu_m \longrightarrow \mu_m^{\underline{\omega}}
\qquad \text{hom.\ diagonal} \\
\mu_m(\underline{\omega}) \simeq D^{+} \longrightarrow
\mu_m^{\underline{\omega}} \\
\qquad \lambda \longmapsto (\lambda \wedge \omega)_{\omega \in
\underline{\omega}}
\end{cases}
\]\[(34)\qquad \text{\struck{\ill{}}}\;
\underbrace{D^{+}}_{\mu_m(\underline{\omega})} \times \mu_m
\longrightarrow \mu_m^{\underline{\omega}}\]
LaTeX source
\[
(34)\qquad \text{\struck{\ill{}}}\;
\underbrace{D^{+}}_{\mu_m(\underline{\omega})} \times \mu_m
\longrightarrow \mu_m^{\underline{\omega}}
\]\[(35)\qquad \text{\struck{\ill{}}}\;
\Bigl(\prod_{\omega \in \underline{\omega}} \lambda_\omega\Bigr)^{n} = 1
\qquad \text{i.e.}\quad (\lambda_\omega \lambda_{\omega'})^{n} = 1 .\]
LaTeX source
\[
(35)\qquad \text{\struck{\ill{}}}\;
\Bigl(\prod_{\omega \in \underline{\omega}} \lambda_\omega\Bigr)^{n} = 1
\qquad \text{i.e.}\quad (\lambda_\omega \lambda_{\omega'})^{n} = 1 .
\]\[(36)\qquad D^{+} \times \mathbb{C}^{*} \hookrightarrow
\mathbb{C}^{*\,\underline{\omega}}\]
LaTeX source
\[
(36)\qquad D^{+} \times \mathbb{C}^{*} \hookrightarrow
\mathbb{C}^{*\,\underline{\omega}}
\]\[(37)\qquad \text{\struck{\ill{}}}\;
(\lambda_\omega \lambda_{\omega'}^{-1})^{n} = 1\]
LaTeX source
\[
(37)\qquad \text{\struck{\ill{}}}\;
(\lambda_\omega \lambda_{\omega'}^{-1})^{n} = 1
\]\[(D \times \mu_m)/\{\pm 1\} \times \{1, \tau\}\]
LaTeX source
\[
(D \times \mu_m)/\{\pm 1\} \times \{1, \tau\}
\]\[\begin{array}{l}
\text{\struck{$D_0 \to D_1$ \quad cinq façons}} \\
D_1 \to D_2 \\
D_{01} \to D_0 \\
D_{01} \to D_1 \\
D_{02} \to D_0 \\
D_{02} \to D_2 \quad \text{\struck{\ill{}}}
\end{array}\]
LaTeX source
\[
\begin{array}{l}
\text{\struck{$D_0 \to D_1$ \quad cinq façons}} \\
D_1 \to D_2 \\
D_{01} \to D_0 \\
D_{01} \to D_1 \\
D_{02} \to D_0 \\
D_{02} \to D_2 \quad \text{\struck{\ill{}}}
\end{array}
\]\[\mathbb{Z}/5\mathbb{Z} \;\to\; \mathbb{D}_5\]
LaTeX source
\[
\mathbb{Z}/5\mathbb{Z} \;\to\; \mathbb{D}_5
\]\[\mathbb{Z}/3\mathbb{Z} \;\to\; \mathfrak{S}_3\]
LaTeX source
\[
\mathbb{Z}/3\mathbb{Z} \;\to\; \mathfrak{S}_3
\]\[\mathcal{U} = X \setminus \{x\} = \{\eta\} \tag{53}\]
LaTeX source
\[
\mathcal{U} = X \setminus \{x\} = \{\eta\} \tag{53}
\]\[T_x = \mathbb{V}(\mathfrak{m}_x/\mathfrak{m}_x^2) = \check{\mathbb{V}}\bigl((\mathfrak{m}_x/\mathfrak{m}_x^2)\bigr) \tag{54}\]
LaTeX source
\[
T_x = \mathbb{V}(\mathfrak{m}_x/\mathfrak{m}_x^2) = \check{\mathbb{V}}\bigl((\mathfrak{m}_x/\mathfrak{m}_x^2)\bigr) \tag{54}
\]\[\hat{\Pi}_1(T_x^*) \longrightarrow \hat{\Pi}_1(\mathcal{U}) \tag{55}\]
LaTeX source
\[
\hat{\Pi}_1(T_x^*) \longrightarrow \hat{\Pi}_1(\mathcal{U}) \tag{55}
\]\[g_1(z) = \psi\bigl(\varphi(z)^2\bigr),\]
LaTeX source
\[ g_1(z) = \psi\bigl(\varphi(z)^2\bigr), \]
\[0 \;\; 1 \;\; \infty \longrightarrow 0 \;\; \infty \;\; 1 ,\]
LaTeX source
\[ 0 \;\; 1 \;\; \infty \longrightarrow 0 \;\; \infty \;\; 1 , \]
\[1 \mapsto 0, \quad -1 \mapsto \infty, \quad 0 \ (\text{ou } \infty) \mapsto 1 ,\]
LaTeX source
\[
1 \mapsto 0, \quad -1 \mapsto \infty, \quad 0 \ (\text{ou } \infty) \mapsto 1 ,
\]\[\psi(z) = \sigma_0(z) = \frac{z}{z-1},\]
LaTeX source
\[
\psi(z) = \sigma_0(z) = \frac{z}{z-1},
\]\[\left\{
\begin{aligned}
& g_1 : \mathbb{X}_1 \longrightarrow \mathbb{X}_0 : \\
& g_1(z) = -\frac{1}{4}\,\frac{(z-1)^2}{z}
\end{aligned}
\right. \tag{26}\]
LaTeX source
\[
\left\{
\begin{aligned}
& g_1 : \mathbb{X}_1 \longrightarrow \mathbb{X}_0 : \\
& g_1(z) = -\frac{1}{4}\,\frac{(z-1)^2}{z}
\end{aligned}
\right. \tag{26}
\]\[\left\{
\begin{aligned}
& g_2 : \mathbb{X}_2 \longrightarrow \mathbb{X}_0 : \\
& g_2(z) = g_1(z^2) = -\frac{1}{4}\,\frac{(z^2-1)^2}{z^2} = -\frac{1}{4}\Bigl(z - \frac{1}{z}\Bigr)^2
\end{aligned}
\right. \tag{27}\]
LaTeX source
\[
\left\{
\begin{aligned}
& g_2 : \mathbb{X}_2 \longrightarrow \mathbb{X}_0 : \\
& g_2(z) = g_1(z^2) = -\frac{1}{4}\,\frac{(z^2-1)^2}{z^2} = -\frac{1}{4}\Bigl(z - \frac{1}{z}\Bigr)^2
\end{aligned}
\right. \tag{27}
\]\[\begin{array}{cc|c|c||cc|cc}
\mathbb{D}_5 & \mathbb{D}_5 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/5\mathbb{Z} & \mathbb{D}_5 & 12 & 12 & 12 & 6 \\
\mathbb{Z}/2\mathbb{Z} & \mathbb{D}_2 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} & 30 & 15 & 30 & 15 \\
\mathbb{Z}/3\mathbb{Z} & \mathfrak{S}_3 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/3\mathbb{Z} & \mathfrak{S}_3 & 20 & 10 & 20 & 10
\end{array}\]
LaTeX source
\[
\begin{array}{cc|c|c||cc|cc}
\mathbb{D}_5 & \mathbb{D}_5 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/5\mathbb{Z} & \mathbb{D}_5 & 12 & 12 & 12 & 6 \\
\mathbb{Z}/2\mathbb{Z} & \mathbb{D}_2 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} & 30 & 15 & 30 & 15 \\
\mathbb{Z}/3\mathbb{Z} & \mathfrak{S}_3 \times \mathbb{Z}/2\mathbb{Z} & \mathbb{Z}/3\mathbb{Z} & \mathfrak{S}_3 & 20 & 10 & 20 & 10
\end{array}
\]\[\begin{array}{ccc}
X_2 & X_1 & X_0 \\
0 & 0 \,|\, \infty & \infty \\
-1 & 1 & 0 \\
\infty & \text{\struck{$-1$}}\ \ast & 1
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
X_2 & X_1 & X_0 \\
0 & 0 \,|\, \infty & \infty \\
-1 & 1 & 0 \\
\infty & \text{\struck{$-1$}}\ \ast & 1
\end{array}
\]\[\frac{(1+z)^2 - (1-z)^2}{(1-z)^2} \Big/ \frac{(1+z)^2}{(1-z)^2}
= \frac{1 - \left(\frac{1+z}{1-z}\right)^2}{\left(\frac{1+z}{1-z}\right)^2},
\qquad = \frac{-4z}{(1-z)^2}\]
LaTeX source
\[
\frac{(1+z)^2 - (1-z)^2}{(1-z)^2} \Big/ \frac{(1+z)^2}{(1-z)^2}
= \frac{1 - \left(\frac{1+z}{1-z}\right)^2}{\left(\frac{1+z}{1-z}\right)^2},
\qquad = \frac{-4z}{(1-z)^2}
\]\[-\frac{1}{4i}\,(i-1)^2 = \frac{1}{2}, \qquad (i-1)^2 = -1+1-2i .\]
LaTeX source
\[
-\frac{1}{4i}\,(i-1)^2 = \frac{1}{2}, \qquad (i-1)^2 = -1+1-2i .
\]\[\sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = (\sigma_0\sigma_\infty)^2 = 1\]
LaTeX source
\[ \sigma_0^2 = \sigma_1^2 = \sigma_\infty^2 = (\sigma_0\sigma_\infty)^2 = 1 \]
\[(\rho_0, \rho_1, \rho_\infty \,;\ \rho_\infty \rho_1 \rho_0 = \rho_1^2 = 1)\]
LaTeX source
\[ (\rho_0, \rho_1, \rho_\infty \,;\ \rho_\infty \rho_1 \rho_0 = \rho_1^2 = 1) \]
\[\begin{array}{lcl}
0 \mapsto 1, & & 1 \mapsto 1, \\
1 \mapsto 1, & -1 \mapsto 0, & 0 \mapsto \infty, \\
i, -i \longmapsto \tfrac{1}{2} & &
\end{array}
\qquad
\varphi(z) = \frac{z-1}{z+1}, \quad \Bigl(\frac{z-1}{z+1}\Bigr)^2 .\]
LaTeX source
\[
\begin{array}{lcl}
0 \mapsto 1, & & 1 \mapsto 1, \\
1 \mapsto 1, & -1 \mapsto 0, & 0 \mapsto \infty, \\
i, -i \longmapsto \tfrac{1}{2} & &
\end{array}
\qquad
\varphi(z) = \frac{z-1}{z+1}, \quad \Bigl(\frac{z-1}{z+1}\Bigr)^2 .
\]\[(0, 1, \infty) \longrightarrow (0, 1, \infty) \longmapsto 0 \;\; \infty \;\; 1
\qquad\qquad 1 \;\; {-1} \;\; 0 .\]
LaTeX source
\[
(0, 1, \infty) \longrightarrow (0, 1, \infty) \longmapsto 0 \;\; \infty \;\; 1
\qquad\qquad 1 \;\; {-1} \;\; 0 .
\]\[\frac{\left(\frac{z-1}{z+1}\right)^2}{\ \cdots\ } = \frac{(z-1)^2}{(z-1)^2 - (z+1)^2}\]
LaTeX source
\[
\frac{\left(\frac{z-1}{z+1}\right)^2}{\ \cdots\ } = \frac{(z-1)^2}{(z-1)^2 - (z+1)^2}
\]\[\hat{M}'_{0,5} \simeq \hat{\mathcal{X}}^*_{0,4} .\]
LaTeX source
\[
\hat{M}'_{0,5} \simeq \hat{\mathcal{X}}^*_{0,4} .
\]\[\hat{\mathcal{X}}_{0,4} \longrightarrow \mathbb{P}^1 \times \mathbb{P}^1\]
LaTeX source
\[
\hat{\mathcal{X}}_{0,4} \longrightarrow \mathbb{P}^1 \times \mathbb{P}^1
\]\[\hat{\mathcal{X}}^*_{0,4} \simeq \hat{\mathcal{X}}_{0,4} - \text{les quatre sections } \tilde{\Delta}_0, \tilde{\Delta}_1, \tilde{\Delta}_\infty, \tilde{\Delta} .\]
LaTeX source
\[
\hat{\mathcal{X}}^*_{0,4} \simeq \hat{\mathcal{X}}_{0,4} - \text{les quatre sections } \tilde{\Delta}_0, \tilde{\Delta}_1, \tilde{\Delta}_\infty, \tilde{\Delta} .
\]\[\hat{M}_{0,3} \;\text{---}\; \hat{\mathcal{X}}_{0,3}\]
LaTeX source
\[
\hat{M}_{0,3} \;\text{---}\; \hat{\mathcal{X}}_{0,3}
\]\[\bigotimes_{i \in K} L_i = L_K\]
LaTeX source
\[
\bigotimes_{i \in K} L_i = L_K
\]\[L_i \simeq L_K \otimes p^*_{K \setminus \{i\}, K}\, L_{K \setminus \{i\}}\]
LaTeX source
\[
L_i \simeq L_K \otimes p^*_{K \setminus \{i\}, K}\, L_{K \setminus \{i\}}
\]\[\begin{array}{l|c|c|c}
n & 4 & 5 & 6 \\ \hline
(n-1)(n-2) & 6 & 12 & 20 \\ \hline
\text{meilleur exposant pour trivialiser } L & 3 & 12\,? & 20\,? \\ \hline
\text{meilleur exposant pour trivialiser } L_i & 6 & 12\,? & 60\,? \\ \hline
\text{exposant pour trivialiser } \text{\struck{$H^1(M_{0,I}, \mathfrak{S}_I;\, \mathbb{G}_m)$}} & & & \\ \hline
\text{dividende (?)} \ H^1(M_{0,I}, \mathfrak{S}_{I \setminus \{i\}};\, \mathbb{G}_m) & & &
\end{array}\]
LaTeX source
\[
\begin{array}{l|c|c|c}
n & 4 & 5 & 6 \\ \hline
(n-1)(n-2) & 6 & 12 & 20 \\ \hline
\text{meilleur exposant pour trivialiser } L & 3 & 12\,? & 20\,? \\ \hline
\text{meilleur exposant pour trivialiser } L_i & 6 & 12\,? & 60\,? \\ \hline
\text{exposant pour trivialiser } \text{\struck{$H^1(M_{0,I}, \mathfrak{S}_I;\, \mathbb{G}_m)$}} & & & \\ \hline
\text{dividende (?)} \ H^1(M_{0,I}, \mathfrak{S}_{I \setminus \{i\}};\, \mathbb{G}_m) & & &
\end{array}
\]\[\cdots\ \prod_{A \in \mathbb{P}_3}{}' p_A^*(L_A)
= \prod_{\substack{(i \in A) \\ i \in A \in \mathbb{P}_3(K)}} L_i
= \Bigl(\prod_{i \in K} L_i\Bigr)^{\otimes 4}
= L^{4}\]
LaTeX source
\[
\cdots\ \prod_{A \in \mathbb{P}_3}{}' p_A^*(L_A)
= \prod_{\substack{(i \in A) \\ i \in A \in \mathbb{P}_3(K)}} L_i
= \Bigl(\prod_{i \in K} L_i\Bigr)^{\otimes 4}
= L^{4}
\]\[(L^4)^3 \simeq \text{\struck{$\mathcal{O}$ \ldots}}\]
LaTeX source
\[
(L^4)^3 \simeq \text{\struck{$\mathcal{O}$ \ldots}}
\]\[\xi_A \in \Gamma(M_{0,I}, L_A^*) ,
\qquad
\prod_A \xi_A \in \Gamma\Bigl(M_{0,I}, \bigotimes_A L_A\Bigr)\]
LaTeX source
\[
\xi_A \in \Gamma(M_{0,I}, L_A^*) ,
\qquad
\prod_A \xi_A \in \Gamma\Bigl(M_{0,I}, \bigotimes_A L_A\Bigr)
\]\[\xi, \quad A \in \mathbb{P}_3(I), \qquad M_{0,I} \xrightarrow{\ \ast\ } M_{0,A}\]
LaTeX source
\[
\xi, \quad A \in \mathbb{P}_3(I), \qquad M_{0,I} \xrightarrow{\ \ast\ } M_{0,A}
\]\[\xi = \prod \xi_A \in L^{\otimes \nu}, \qquad \tfrac{(n-1)(n-2)}{2} = \nu\]
LaTeX source
\[
\xi = \prod \xi_A \in L^{\otimes \nu}, \qquad \tfrac{(n-1)(n-2)}{2} = \nu
\]\[L_i \xrightarrow{\ \varphi_{i,A,\omega}\ } \mathcal{O}, \qquad i \in A \in \mathbb{P}_3(I),
\qquad
L = \bigotimes_{i \in I} L_i ,
\qquad
\nu = \frac{(n-1)(n-2)}{2}\]
LaTeX source
\[
L_i \xrightarrow{\ \varphi_{i,A,\omega}\ } \mathcal{O}, \qquad i \in A \in \mathbb{P}_3(I),
\qquad
L = \bigotimes_{i \in I} L_i ,
\qquad
\nu = \frac{(n-1)(n-2)}{2}
\]\[L_i^2 \xrightarrow[\ \sim\ ]{\ \psi_{i,A}\ } \mathcal{O}
\qquad
\text{\struck{$L_i \xrightarrow{\ \sim\ } \mathcal{O}$}}
\qquad
\xi \in L_i \ \text{---}\ \xi\]
LaTeX source
\[
L_i^2 \xrightarrow[\ \sim\ ]{\ \psi_{i,A}\ } \mathcal{O}
\qquad
\text{\struck{$L_i \xrightarrow{\ \sim\ } \mathcal{O}$}}
\qquad
\xi \in L_i \ \text{---}\ \xi
\]\[\xi^*_{i,A} \in \Gamma L_i^2 \qquad \xi \in L_A^2 \simeq\]
LaTeX source
\[
\xi^*_{i,A} \in \Gamma L_i^2 \qquad \xi \in L_A^2 \simeq
\]\[L_i \simeq L \otimes L^{-1}_{I \setminus \{i\}} \simeq L \otimes p_i^*\bigl(\mathcal{O}(\ast)\bigr),
\qquad
p_i : M_{0,I} \longrightarrow M_{0, I \setminus \{i\}} \simeq \Sigma\]
LaTeX source
\[
L_i \simeq L \otimes L^{-1}_{I \setminus \{i\}} \simeq L \otimes p_i^*\bigl(\mathcal{O}(\ast)\bigr),
\qquad
p_i : M_{0,I} \longrightarrow M_{0, I \setminus \{i\}} \simeq \Sigma
\]\[\mathcal{O}(4)^{\ast}\ \cdots \simeq \mathcal{O}(-2) \simeq \Omega^1\]
LaTeX source
\[
\mathcal{O}(4)^{\ast}\ \cdots \simeq \mathcal{O}(-2) \simeq \Omega^1
\]\[H \qquad \mathcal{O}( \qquad \prod_{i \in I} L_i\]
LaTeX source
\[
H \qquad \mathcal{O}( \qquad \prod_{i \in I} L_i
\]\[M_{0,I} \hookrightarrow \prod_{i \in K} \Sigma_{J(i)}, \qquad J(i) = I \setminus \{i\}\]
LaTeX source
\[
M_{0,I} \hookrightarrow \prod_{i \in K} \Sigma_{J(i)}, \qquad J(i) = I \setminus \{i\}
\]\[H^1(\mathfrak{S}_5, M^!_{0,5};\, \mathbb{G}_m), \qquad H^1(M_{0,5};\, \mathbb{G}_m) \simeq 0,
\qquad H^0(M^!_{0,5}, \mathbb{G}_m),\]
LaTeX source
\[
H^1(\mathfrak{S}_5, M^!_{0,5};\, \mathbb{G}_m), \qquad H^1(M_{0,5};\, \mathbb{G}_m) \simeq 0,
\qquad H^0(M^!_{0,5}, \mathbb{G}_m),
\]\[H^2(\mathcal{T}^+_{0,5}, \mathbb{Z}) = \qquad H^1(M^!_{0,5}\]
LaTeX source
\[
H^2(\mathcal{T}^+_{0,5}, \mathbb{Z}) = \qquad H^1(M^!_{0,5}
\]\[H^0(M_{0,3}, \mathbb{G}_m) \simeq \text{\struck{$\ast$}}\]
LaTeX source
\[
H^0(M_{0,3}, \mathbb{G}_m) \simeq \text{\struck{$\ast$}}
\]\[0 \longrightarrow \mathbb{G}_m \longrightarrow H^0(M_{0,K}, \mathbb{G}_m) \longrightarrow \bigl(\mathbb{Z}^{\mathbb{P}_3(K)}\bigr)' \longrightarrow 0\]
LaTeX source
\[
0 \longrightarrow \mathbb{G}_m \longrightarrow H^0(M_{0,K}, \mathbb{G}_m) \longrightarrow \bigl(\mathbb{Z}^{\mathbb{P}_3(K)}\bigr)' \longrightarrow 0
\]\[0 \longrightarrow \underbrace{H^1(\mathfrak{S}_K, \mathbb{G}_m)}_{\pm 1}
\longrightarrow \underbrace{H^1\bigl(\mathfrak{S}_K, H^0(M_{0,K}, \mathbb{G}_m)\bigr)}_{H^1(M_{0,K}, \mathfrak{S}_K;\, \mathbb{G}_m)}
\longrightarrow H^1\bigl(\mathfrak{S}_5, (\mathbb{Z}^{\mathbb{P}_3(K)})'\bigr)\]
LaTeX source
\[
0 \longrightarrow \underbrace{H^1(\mathfrak{S}_K, \mathbb{G}_m)}_{\pm 1}
\longrightarrow \underbrace{H^1\bigl(\mathfrak{S}_K, H^0(M_{0,K}, \mathbb{G}_m)\bigr)}_{H^1(M_{0,K}, \mathfrak{S}_K;\, \mathbb{G}_m)}
\longrightarrow H^1\bigl(\mathfrak{S}_5, (\mathbb{Z}^{\mathbb{P}_3(K)})'\bigr)
\]\[H^1\bigl(\mathfrak{S}_5, (\mathbb{Z}^{\mathbb{P}_3(K)})'\bigr) \longrightarrow H^2(\mathfrak{S}_K, \mathbb{G}_m)\]
LaTeX source
\[
H^1\bigl(\mathfrak{S}_5, (\mathbb{Z}^{\mathbb{P}_3(K)})'\bigr) \longrightarrow H^2(\mathfrak{S}_K, \mathbb{G}_m)
\]\[\boxed{\rho_1 \longleftrightarrow \sigma_\infty, \quad \rho_\infty \longmapsto \rho^{-1}}
\qquad
\rho_0 = (\rho_\infty \rho_1)^{-1} \longrightarrow \sigma_\infty \rho^{\ast} = \varepsilon_0
\tag{41}\]
LaTeX source
\[
\boxed{\rho_1 \longleftrightarrow \sigma_\infty, \quad \rho_\infty \longmapsto \rho^{-1}}
\qquad
\rho_0 = (\rho_\infty \rho_1)^{-1} \longrightarrow \sigma_\infty \rho^{\ast} = \varepsilon_0
\tag{41}
\]\[\boxed{\tau_0 \longmapsto \tau_\infty}, \quad
\underset{\displaystyle \rho_0}{\tau_0 \,\shortparallel} \longmapsto \quad , \quad
\boxed{\tau_\infty \longmapsto \tau_\infty} ,
\tag{42}\]
LaTeX source
\[
\boxed{\tau_0 \longmapsto \tau_\infty}, \quad
\underset{\displaystyle \rho_0}{\tau_0 \,\shortparallel} \longmapsto \quad , \quad
\boxed{\tau_\infty \longmapsto \tau_\infty} ,
\tag{42}
\]\[\pi_{\mathbb{D}_n}, \quad \pi_{\mathfrak{S}_4^+}, \quad \pi_{\mathfrak{S}_4}, \quad \pi_{\mathfrak{S}_5^+}\]
LaTeX source
\[
\pi_{\mathbb{D}_n}, \quad \pi_{\mathfrak{S}_4^+}, \quad \pi_{\mathfrak{S}_4}, \quad \pi_{\mathfrak{S}_5^+}
\]\[2 \times (6 + 15 + 10) = 62\]
LaTeX source
\[ 2 \times (6 + 15 + 10) = 62 \]
\[1 \to L \longrightarrow \tilde{\pi} \longrightarrow \pi \longrightarrow 1\]
LaTeX source
\[
1 \to L \longrightarrow \tilde{\pi} \longrightarrow \pi \longrightarrow 1
\]\[R_\alpha\bigl((x_i)\bigr) = 1 \quad (\alpha \in A)\]
LaTeX source
\[ R_\alpha\bigl((x_i)\bigr) = 1 \quad (\alpha \in A) \]
\[S_\beta\bigl((a_j)\bigr) = 1 \quad (\beta \in B)\]
LaTeX source
\[ S_\beta\bigl((a_j)\bigr) = 1 \quad (\beta \in B) \]
\[\left\{
\begin{aligned}
& R_\alpha\bigl((\tilde{x}_i)\bigr) = M_\alpha\bigl((a_j)\bigr) && \forall \alpha \in A \qquad (1) \\
& \tilde{x}_{i_0}\, a_{j_0}\, \tilde{x}_{i_0}^{-1} = M_{i_0 j_0}\bigl((a_j)\bigr) && \forall (i_0, j_0) \in I \times J \qquad (2)
\end{aligned}
\right.\]
LaTeX source
\[
\left\{
\begin{aligned}
& R_\alpha\bigl((\tilde{x}_i)\bigr) = M_\alpha\bigl((a_j)\bigr) && \forall \alpha \in A \qquad (1) \\
& \tilde{x}_{i_0}\, a_{j_0}\, \tilde{x}_{i_0}^{-1} = M_{i_0 j_0}\bigl((a_j)\bigr) && \forall (i_0, j_0) \in I \times J \qquad (2)
\end{aligned}
\right.
\]\[S_\beta\bigl((a_j)\bigr) = 1 \quad (\beta \in B) \qquad (3)\]
LaTeX source
\[ S_\beta\bigl((a_j)\bigr) = 1 \quad (\beta \in B) \qquad (3) \]
\[\tilde{x}_i\, a_j\, \tilde{x}_i^{-1} = a_j, \quad \text{i.e.}\ [\tilde{x}_i, a_j] = 1 \qquad (2')\]
LaTeX source
\[
\tilde{x}_i\, a_j\, \tilde{x}_i^{-1} = a_j, \quad \text{i.e.}\ [\tilde{x}_i, a_j] = 1 \qquad (2')
\]\[\widetilde{\mathcal{T}} \xrightarrow{\;p\;} \mathcal{T}\]
LaTeX source
\[
\widetilde{\mathcal{T}} \xrightarrow{\;p\;} \mathcal{T}
\]\[(4) \qquad u'_{i'^{(\alpha)}_{n_\alpha}} \cdots u'_{i'^{(\alpha)}_{0}} = 1
\quad \text{dans } \mathcal{T}\]
LaTeX source
\[
(4) \qquad u'_{i'^{(\alpha)}_{n_\alpha}} \cdots u'_{i'^{(\alpha)}_{0}} = 1
\quad \text{dans } \mathcal{T}
\]\[(5) \qquad a'_{j'^{(\beta)}_{n_\beta}}\, a'_{j'^{(\beta)}_{n_\beta - 1}} \cdots a'_{j'^{(\beta)}_0} = 1\]
LaTeX source
\[
(5) \qquad a'_{j'^{(\beta)}_{n_\beta}}\, a'_{j'^{(\beta)}_{n_\beta - 1}} \cdots a'_{j'^{(\beta)}_0} = 1
\]\[(1) \qquad \widetilde{\mathcal{T}} \longrightarrow \mathcal{T}\]
LaTeX source
\[
(1) \qquad \widetilde{\mathcal{T}} \longrightarrow \mathcal{T}
\]\[(2) \qquad (\tilde{u}_i)_{i \in I}\]
LaTeX source
\[
(2) \qquad (\tilde{u}_i)_{i \in I}
\]\[(3) \qquad (u_i)_{i \in I}\]
LaTeX source
\[
(3) \qquad (u_i)_{i \in I}
\]\[(4) \qquad I' = I \times \{\pm 1\}\]
LaTeX source
\[
(4) \qquad I' = I \times \{\pm 1\}
\]\[(5) \qquad u'_{i'} = u_i^{\varepsilon}\]
LaTeX source
\[
(5) \qquad u'_{i'} = u_i^{\varepsilon}
\]\[(6) \qquad \hat{I}' = \coprod_{n \in \mathbf{N}^*} I'^{\,n} .\]
LaTeX source
\[
(6) \qquad \hat{I}' = \coprod_{n \in \mathbf{N}^*} I'^{\,n} .
\]\[(7) \qquad i'^{(\alpha)} \in I'^{\,n_\alpha}, \qquad
i'^{(\alpha)} = (i'_{\alpha,1}, i'_{\alpha,2}, \ldots, i'_{\alpha,n_\alpha})\]
LaTeX source
\[
(7) \qquad i'^{(\alpha)} \in I'^{\,n_\alpha}, \qquad
i'^{(\alpha)} = (i'_{\alpha,1}, i'_{\alpha,2}, \ldots, i'_{\alpha,n_\alpha})
\]\[u'_{i'_{\alpha,1}}, u'_{i'_{\alpha,2}}, \ldots, u'_{i'_{\alpha,n_\alpha}} \in \operatorname{Fl}\mathcal{T}\]
LaTeX source
\[
u'_{i'_{\alpha,1}}, u'_{i'_{\alpha,2}}, \ldots, u'_{i'_{\alpha,n_\alpha}} \in \operatorname{Fl}\mathcal{T}
\]\[(8) \qquad \underbrace{u'_{i'_{\alpha,1}}\, u'_{i'_{\alpha,2}} \cdots u'_{i'_{\alpha,n_\alpha}}}_{M_\alpha((u_i)_{i \in I})} = 1 .\]
LaTeX source
\[
(8) \qquad \underbrace{u'_{i'_{\alpha,1}}\, u'_{i'_{\alpha,2}} \cdots u'_{i'_{\alpha,n_\alpha}}}_{M_\alpha((u_i)_{i \in I})} = 1 .
\]\[u'_{i'_1}\, u'_{i'_2} \cdots u'_{i'_n} = 1\]
LaTeX source
\[
u'_{i'_1}\, u'_{i'_2} \cdots u'_{i'_n} = 1
\]\[(9) \qquad
\tilde{s}_{i'_n} \to \tilde{t}_{i'_n} \quad
\tilde{s}_{i'_{n-1}} \to \tilde{t}_{i'_{n-1}} \quad \cdots \quad
\tilde{s}_{i'_2} \to \tilde{t}_{i'_2} \quad
\tilde{s}_{i'_1} \to \tilde{t}_{i'_1}\]
LaTeX source
\[
(9) \qquad
\tilde{s}_{i'_n} \to \tilde{t}_{i'_n} \quad
\tilde{s}_{i'_{n-1}} \to \tilde{t}_{i'_{n-1}} \quad \cdots \quad
\tilde{s}_{i'_2} \to \tilde{t}_{i'_2} \quad
\tilde{s}_{i'_1} \to \tilde{t}_{i'_1}
\]\[(10) \qquad (a_j)_{j \in J}, \qquad a_j \in \operatorname{Fl}(\widetilde{\mathcal{T}})\]
LaTeX source
\[
(10) \qquad (a_j)_{j \in J}, \qquad a_j \in \operatorname{Fl}(\widetilde{\mathcal{T}})
\]\[M_{\alpha,1}((a_j)) : \tilde{t}_{i'_1} \rightsquigarrow \tilde{s}_{i'_n}\]
LaTeX source
\[
M_{\alpha,1}((a_j)) : \tilde{t}_{i'_1} \rightsquigarrow \tilde{s}_{i'_n}
\]\[(12) \qquad
\boxed{\begin{gathered}
M_{\alpha,1}((a_j))\, u'_{i'_{\alpha,1}}\, M_{\alpha,2}((a_j))\, u'_{i'_{\alpha,2}} \cdots \\
\cdots M_{\alpha,i'_{\alpha,n_\alpha}}((a_j))\, u'_{i'_{\alpha,n_\alpha}} = 1
\end{gathered}}\]
LaTeX source
\[
(12) \qquad
\boxed{\begin{gathered}
M_{\alpha,1}((a_j))\, u'_{i'_{\alpha,1}}\, M_{\alpha,2}((a_j))\, u'_{i'_{\alpha,2}} \cdots \\
\cdots M_{\alpha,i'_{\alpha,n_\alpha}}((a_j))\, u'_{i'_{\alpha,n_\alpha}} = 1
\end{gathered}}
\]\[(13) \qquad \boxed{N'_\beta((a_j)) = 1}\]
LaTeX source
\[
(13) \qquad \boxed{N'_\beta((a_j)) = 1}
\]\[(14) \qquad
\boxed{\underbrace{\tilde{u}_i(a_{j_0})}_{\substack{\text{flèche de la}\\ \text{catégorie fibre}\\ \text{contenant ext.}(\tilde{u}_i)}}
= M''_{i,j}((a_j))} .\]
LaTeX source
\[
(14) \qquad
\boxed{\underbrace{\tilde{u}_i(a_{j_0})}_{\substack{\text{flèche de la}\\ \text{catégorie fibre}\\ \text{contenant ext.}(\tilde{u}_i)}}
= M''_{i,j}((a_j))} .
\]\[(1) \qquad f : C' \longrightarrow C\]
LaTeX source
\[ (1) \qquad f : C' \longrightarrow C \]
\[(2) \qquad \pi \longrightarrow \operatorname{End}(\mathrm{id}_{C'})\]
LaTeX source
\[
(2) \qquad \pi \longrightarrow \operatorname{End}(\mathrm{id}_{C'})
\]\[1 \longrightarrow \pi \longrightarrow \operatorname{Aut}_{C'}(x) \longrightarrow
\operatorname{Aut}_{C'}(f(x)) \longrightarrow 1 .\]
LaTeX source
\[
1 \longrightarrow \pi \longrightarrow \operatorname{Aut}_{C'}(x) \longrightarrow
\operatorname{Aut}_{C'}(f(x)) \longrightarrow 1 .
\]\[\pi_0(f) : \pi_0(C) \xrightarrow{\;\sim\;} \pi_0(C') .\]
LaTeX source
\[
\pi_0(f) : \pi_0(C) \xrightarrow{\;\sim\;} \pi_0(C') .
\]\[\text{\struck{$\pi_1(P_i, x_i) \to$}}\]
LaTeX source
\[
\text{\struck{$\pi_1(P_i, x_i) \to$}}
\]\[\text{\struck{\ill{}}}\quad \pi \longrightarrow \pi_1(P_i, x_i) \longrightarrow \pi_1(T_i, t_i) \longrightarrow 1\]
LaTeX source
\[
\text{\struck{\ill{}}}\quad \pi \longrightarrow \pi_1(P_i, x_i) \longrightarrow \pi_1(T_i, t_i) \longrightarrow 1
\]\[\Pi_1(p) : \Pi_1(P) \longrightarrow \Pi_1(T)\]
LaTeX source
\[ \Pi_1(p) : \Pi_1(P) \longrightarrow \Pi_1(T) \]
\[(6) \qquad \mu \xrightarrow{\;i_\mu\;} U\]
LaTeX source
\[
(6) \qquad \mu \xrightarrow{\;i_\mu\;} U
\]\[(7) \qquad E \xrightarrow{\;i_E\;} P\]
LaTeX source
\[
(7) \qquad E \xrightarrow{\;i_E\;} P
\]\[(8) \qquad \operatorname{Hom}_{C'}(x, y) = \operatorname{Hom}_{\Pi_1(P)}(i_E(x), i_E(y))
\qquad x, y \in E\]
LaTeX source
\[
(8) \qquad \operatorname{Hom}_{C'}(x, y) = \operatorname{Hom}_{\Pi_1(P)}(i_E(x), i_E(y))
\qquad x, y \in E
\]\[C' \longrightarrow \Pi_1(P)\]
LaTeX source
\[ C' \longrightarrow \Pi_1(P) \]
\[(9) \qquad E_0 = E/\mu \xrightarrow{\;i_{E_0}\;} T = P/U\]
LaTeX source
\[
(9) \qquad E_0 = E/\mu \xrightarrow{\;i_{E_0}\;} T = P/U
\]\[(10) \qquad \operatorname{Hom}_C(x_0, y_0) = \operatorname{Hom}_{\Pi_1(T)}(i_{E_0}(x_0), i_{E_0}(y_0)) .\]
LaTeX source
\[
(10) \qquad \operatorname{Hom}_C(x_0, y_0) = \operatorname{Hom}_{\Pi_1(T)}(i_{E_0}(x_0), i_{E_0}(y_0)) .
\]\[(12) \qquad 0 \longrightarrow \pi \longrightarrow \widetilde{U} \longrightarrow U \longrightarrow 0\]
LaTeX source
\[
(12) \qquad 0 \longrightarrow \pi \longrightarrow \widetilde{U} \longrightarrow U \longrightarrow 0
\]\[(13) \qquad 0 \longrightarrow \pi \longrightarrow \mathcal{E} \longrightarrow \mu \longrightarrow 0
\qquad \text{ext.\ \uncertain{commutative}}\]
LaTeX source
\[
(13) \qquad 0 \longrightarrow \pi \longrightarrow \mathcal{E} \longrightarrow \mu \longrightarrow 0
\qquad \text{ext.\ \uncertain{commutative}}
\]\[(14) \qquad \operatorname{Isom}_{C'_\xi}(x, y) \simeq \mathcal{E}_{y - x}\]
LaTeX source
\[
(14) \qquad \operatorname{Isom}_{C'_\xi}(x, y) \simeq \mathcal{E}_{y - x}
\]\[(15) \qquad C'_\xi \simeq \mathcal{E}(E_\xi)\]
LaTeX source
\[
(15) \qquad C'_\xi \simeq \mathcal{E}(E_\xi)
\]\[(16) \qquad f : C' \longrightarrow C\]
LaTeX source
\[ (16) \qquad f : C' \longrightarrow C \]
\[\pi \longrightarrow \text{\struck{$\operatorname{Aut}$}}\,(\mathrm{id}_{C'})\]
LaTeX source
\[
\pi \longrightarrow \text{\struck{$\operatorname{Aut}$}}\,(\mathrm{id}_{C'})
\]\[H_{x,y} \xrightarrow[\;\sim\;]{\;\alpha_*\;} H_{\alpha x, \alpha y} \qquad \alpha \in \mu\]
LaTeX source
\[
H_{x,y} \xrightarrow[\;\sim\;]{\;\alpha_*\;} H_{\alpha x, \alpha y} \qquad \alpha \in \mu
\]\[H_u \wedge H_v \xrightarrow{\;\sim\;} H_{uv}\]
LaTeX source
\[
H_u \wedge H_v \xrightarrow{\;\sim\;} H_{uv}
\]\[1 \longrightarrow \pi \longrightarrow
\underbrace{\mathcal{E}_\Phi}_{\substack{\Phi \text{ avec sa}\\ (\pi,\mu)\text{ structure}}}
\longrightarrow \mu \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow \pi \longrightarrow
\underbrace{\mathcal{E}_\Phi}_{\substack{\Phi \text{ avec sa}\\ (\pi,\mu)\text{ structure}}}
\longrightarrow \mu \longrightarrow 1
\]\[1 \longrightarrow \pi \longrightarrow \mathcal{E} \longrightarrow \mu \longrightarrow 1\]
LaTeX source
\[
1 \longrightarrow \pi \longrightarrow \mathcal{E} \longrightarrow \mu \longrightarrow 1
\]\[(17) \qquad
\left[\begin{array}{l}
\pi\text{-groupoïdes conn.\ } \Phi\text{, avec}\\
\text{op.\ du groupe } \mu \text{ sur}\\
(\Phi, \pi)\text{, tel que } \operatorname{Ob}\Phi\\
\text{soit un } \mu\text{-torseur}
\end{array}\right]
\xrightarrow{\;\approx\;}
\left[\begin{array}{l}
\text{extensions de}\\
\mu \text{ par } \pi
\end{array}\right]\]
LaTeX source
\[
(17) \qquad
\left[\begin{array}{l}
\pi\text{-groupoïdes conn.\ } \Phi\text{, avec}\\
\text{op.\ du groupe } \mu \text{ sur}\\
(\Phi, \pi)\text{, tel que } \operatorname{Ob}\Phi\\
\text{soit un } \mu\text{-torseur}
\end{array}\right]
\xrightarrow{\;\approx\;}
\left[\begin{array}{l}
\text{extensions de}\\
\mu \text{ par } \pi
\end{array}\right]
\]\[E \longrightarrow P, \qquad \mu \longrightarrow U,\]
LaTeX source
\[ E \longrightarrow P, \qquad \mu \longrightarrow U, \]
\[(19) \qquad U_0 = U/\mu\]
LaTeX source
\[ (19) \qquad U_0 = U/\mu \]
\[(20) \qquad \mathcal{E} \simeq \pi_1(U_0, 1_{U_0}) .\]
LaTeX source
\[
(20) \qquad \mathcal{E} \simeq \pi_1(U_0, 1_{U_0}) .
\]\[(21) \qquad \mathcal{E} \longrightarrow \operatorname{Aut} \mathrm{id}_{\Pi_1(P_0)}\]
LaTeX source
\[
(21) \qquad \mathcal{E} \longrightarrow \operatorname{Aut} \mathrm{id}_{\Pi_1(P_0)}
\]\[\ell \cdot \gamma_{x_0} = {}^{\ell}\gamma_{y_0} \cdot \ell
\qquad \text{où } {}^{\ell}\gamma_{y_0} = \ell \cdot \gamma \cdot \ell^{-1}\]
LaTeX source
\[
\ell \cdot \gamma_{x_0} = {}^{\ell}\gamma_{y_0} \cdot \ell
\qquad \text{où } {}^{\ell}\gamma_{y_0} = \ell \cdot \gamma \cdot \ell^{-1}
\]\[E \longrightarrow P \xrightarrow{\;q\;} P_0\]
LaTeX source
\[
E \longrightarrow P \xrightarrow{\;q\;} P_0
\]\[\operatorname{Ob}(C' \wedge_\pi \mathcal{E}) = \operatorname{Ob} C' \overset{\mathrm{def}}{=} E' ,\]
LaTeX source
\[
\operatorname{Ob}(C' \wedge_\pi \mathcal{E}) = \operatorname{Ob} C' \overset{\mathrm{def}}{=} E' ,
\]\[C_0 \quad \text{\struck{$\longrightarrow$}}\]
LaTeX source
\[
C_0 \quad \text{\struck{$\longrightarrow$}}
\]\[1 \longrightarrow \mathcal{E} \longrightarrow G_0 \longrightarrow G \longrightarrow 1,
\qquad G_0 = \pi_1(P_0), \quad G = \pi_1(T)\]
LaTeX source
\[
1 \longrightarrow \mathcal{E} \longrightarrow G_0 \longrightarrow G \longrightarrow 1,
\qquad G_0 = \pi_1(P_0), \quad G = \pi_1(T)
\]\[\text{i.e.\ on a} \quad \pi_{1,\mathrm{ab}}(C_0) \longrightarrow \mu\]
LaTeX source
\[
\text{i.e.\ on a} \quad \pi_{1,\mathrm{ab}}(C_0) \longrightarrow \mu
\]\[\text{\struck{$\operatorname{Ker}\varphi$}} \qquad
1 \longrightarrow \pi \longrightarrow \operatorname{Ker}\varphi \xrightarrow{\;\mathrm{surj}\;} G \longrightarrow 1\]
LaTeX source
\[
\text{\struck{$\operatorname{Ker}\varphi$}} \qquad
1 \longrightarrow \pi \longrightarrow \operatorname{Ker}\varphi \xrightarrow{\;\mathrm{surj}\;} G \longrightarrow 1
\]\[\pi_1(C_0, x_0) \longrightarrow \mu \simeq \operatorname{Aut}(E_{x_0})\]
LaTeX source
\[
\pi_1(C_0, x_0) \longrightarrow \mu \simeq \operatorname{Aut}(E_{x_0})
\]\[P \xrightarrow{\;f\;} T\]
LaTeX source
\[
P \xrightarrow{\;f\;} T
\]\[f^*(\Pi_T) \text{ \struck{\ill{}} opère sur } \Pi_1 P, \qquad
f^*(\Pi_T) \longrightarrow \operatorname{Int} \Pi_1 P,
\qquad \Pi_1 P \longrightarrow \Pi_1 T\]
LaTeX source
\[
f^*(\Pi_T) \text{ \struck{\ill{}} opère sur } \Pi_1 P, \qquad
f^*(\Pi_T) \longrightarrow \operatorname{Int} \Pi_1 P,
\qquad \Pi_1 P \longrightarrow \Pi_1 T
\]\[U_{0T} = U_T / \gamma_T \qquad \text{syst.\ local des } \pi_1 \text{ est } \mathcal{E}_T\]
LaTeX source
\[
U_{0T} = U_T / \gamma_T \qquad \text{syst.\ local des } \pi_1 \text{ est } \mathcal{E}_T
\]\[0 \longrightarrow \Pi_T \longrightarrow \mathcal{E}_T \longrightarrow \gamma_T \longrightarrow 1\]
LaTeX source
\[
0 \longrightarrow \Pi_T \longrightarrow \mathcal{E}_T \longrightarrow \gamma_T \longrightarrow 1
\]\[S_0\mathcal{T}_G =
\Bigl(\prod_{s \in S}
\underbrace{S_0\mathcal{T}_{0,\hat{A}_s}}_{\substack{(\mathbf{Z}^{\hat{A}_s},\, \mu_2^{\hat{A}_s})\\ \text{-groupoïde}}}
\Bigr)
\wedge^{(\mathbf{Z}^{\hat{A}_s},\, \mu_2^{\hat{A}})}
(\mathbf{Z}^{\hat{A}}, \mu_2^{\hat{A}})\]
LaTeX source
\[
S_0\mathcal{T}_G =
\Bigl(\prod_{s \in S}
\underbrace{S_0\mathcal{T}_{0,\hat{A}_s}}_{\substack{(\mathbf{Z}^{\hat{A}_s},\, \mu_2^{\hat{A}_s})\\ \text{-groupoïde}}}
\Bigr)
\wedge^{(\mathbf{Z}^{\hat{A}_s},\, \mu_2^{\hat{A}})}
(\mathbf{Z}^{\hat{A}}, \mu_2^{\hat{A}})
\]\[R_0\mathcal{T}_G = \prod_{s \in S}
\underbrace{R_0\mathcal{T}_{G_s^b}}_{S_0\mathcal{T}_{G_s^b}}
= \prod_{s \in S} \mathcal{T}_{0,\hat{A}_s}\]
LaTeX source
\[
R_0\mathcal{T}_G = \prod_{s \in S}
\underbrace{R_0\mathcal{T}_{G_s^b}}_{S_0\mathcal{T}_{G_s^b}}
= \prod_{s \in S} \mathcal{T}_{0,\hat{A}_s}
\]\[SP_G \longrightarrow SP_{G^{\natural}}\]
LaTeX source
\[
SP_G \longrightarrow SP_{G^{\natural}}
\]\[P_{G,C} = \prod_{a \in C \,/ M_G} P_{G,a}, \qquad
\mathfrak{T}_{G,C} = \Pi_1 P_{G,\ldots}\]
LaTeX source
\[
P_{G,C} = \prod_{a \in C \,/ M_G} P_{G,a}, \qquad
\mathfrak{T}_{G,C} = \Pi_1 P_{G,\ldots}
\]\[SP_G = P_{G,\hat{A}}, \quad RP_G = P_{G,\emptyset}, \quad SRP_G = P_{G,I}, \quad
P_G \simeq P_{G,A}.\]
LaTeX source
\[
SP_G = P_{G,\hat{A}}, \quad RP_G = P_{G,\emptyset}, \quad SRP_G = P_{G,I}, \quad
P_G \simeq P_{G,A}.
\]\[C \subset A = \mathrm{Sing}(X), \qquad \text{soit } B = A \setminus C .\]
LaTeX source
\[
C \subset A = \mathrm{Sing}(X), \qquad \text{soit } B = A \setminus C .
\]\[X_1 = \widetilde{X} - \bigcup_{x \in \widetilde{C}} D_x^{\circ}\]
LaTeX source
\[
X_1 = \widetilde{X} - \bigcup_{x \in \widetilde{C}} D_x^{\circ}
\]\[\partial X_1 = \coprod_{x \in \widetilde{C}} \partial D_x .\]
LaTeX source
\[
\partial X_1 = \coprod_{x \in \widetilde{C}} \partial D_x .
\]\[\mathbb{U} = \{ z \in \mathbb{C} \mid |z| = 1 \}\]
LaTeX source
\[
\mathbb{U} = \{ z \in \mathbb{C} \mid |z| = 1 \}
\]\[\mathrm{Isom}^{-}_{\mathbb{U}}(T_{x'}, T_{x''}) \simeq
\mathrm{Isom}_{\mathbb{U}}(T_{x'}^{-1}, T_{x''})\]
LaTeX source
\[
\mathrm{Isom}^{-}_{\mathbb{U}}(T_{x'}, T_{x''}) \simeq
\mathrm{Isom}_{\mathbb{U}}(T_{x'}^{-1}, T_{x''})
\]\[T_x = \bigwedge_{\tilde{x} \in C_x} T_{\tilde{x}} .\]
LaTeX source
\[
T_x = \bigwedge_{\tilde{x} \in C_x} T_{\tilde{x}} .
\]\[P_C = P_{C,X} = \prod_{x \in C} T_x\]
LaTeX source
\[
P_C = P_{C,X} = \prod_{x \in C} T_x
\]\[\widetilde{M}^C_G \longrightarrow M_{G'} .\]
LaTeX source
\[
\widetilde{M}^C_G \longrightarrow M_{G'} .
\]\[M^C_G \longrightarrow M_G \quad \text{équiv. d'homotopie}\]
LaTeX source
\[
M^C_G \longrightarrow M_G \quad \text{équiv. d'homotopie}
\]\[T_{X'} = \prod_{x \in \hat{A}'} T_{X',x}\]
LaTeX source
\[
T_{X'} = \prod_{x \in \hat{A}'} T_{X',x}
\]\[S\mathfrak{T}_{G'} = \pi_1(SP_{G'}) .\]
LaTeX source
\[
S\mathfrak{T}_{G'} = \pi_1(SP_{G'}) .
\]\[SP_G = P_C \times_{M_G} P_B
\qquad \text{(isom. de $\mathbb{U}^{\hat{A}}$-torseurs sur $M_G$)}\]
LaTeX source
\[
SP_G = P_C \times_{M_G} P_B
\qquad \text{(isom. de $\mathbb{U}^{\hat{A}}$-torseurs sur $M_G$)}
\]\[S\mathfrak{T}_G \overset{\text{déf}}{=} \pi_1(SP_G) \xleftarrow{\ \sim\ } \pi_1(\widetilde{P}^B_G) .\]
LaTeX source
\[
S\mathfrak{T}_G \overset{\text{déf}}{=} \pi_1(SP_G) \xleftarrow{\ \sim\ } \pi_1(\widetilde{P}^B_G) .
\]\[\mu_{\vec{C}} = \, ]0,1]^{\vec{C}}\]
LaTeX source
\[
\mu_{\vec{C}} = \, ]0,1]^{\vec{C}}
\]\[\Gamma_{i\rho} \cap \Sigma_{\mathbb{R}} \simeq I \setminus \{i\}\]
LaTeX source
\[
\Gamma_{i\rho} \cap \Sigma_{\mathbb{R}} \simeq I \setminus \{i\}
\]\[\Gamma_{i\rho} \simeq \text{\struck{$\Sigma$}}\, \mathbb{U} \wedge_{\{\pm 1\}} (I \setminus \{i\})\]
LaTeX source
\[
\Gamma_{i\rho} \simeq \text{\struck{$\Sigma$}}\, \mathbb{U} \wedge_{\{\pm 1\}} (I \setminus \{i\})
\]\[\widetilde{X}_G = \coprod_{s \in G} \Sigma_{\hat{\vec{A}}_s}\]
LaTeX source
\[
\widetilde{X}_G = \coprod_{s \in G} \Sigma_{\hat{\vec{A}}_s}
\]\[T_x \simeq \mathbb{U} \wedge_{\{\pm 1\}} \{ \hat{\vec{A}}_s \setminus x \}
\quad \text{si } x \in I \simeq \hat{\vec{A}}^{\sigma}\]
LaTeX source
\[
T_x \simeq \mathbb{U} \wedge_{\{\pm 1\}} \{ \hat{\vec{A}}_s \setminus x \}
\quad \text{si } x \in I \simeq \hat{\vec{A}}^{\sigma}
\]\[T_x \simeq \mathbb{U} \wedge_{\{\pm 1\}} \bigwedge_{\tilde{x} \in \hat{\vec{A}}_x}
\bigl( \hat{\vec{A}}_{\sigma(\tilde{x})} \setminus \{\tilde{x}\} \bigr)
\quad \text{si } x \in A .\]
LaTeX source
\[
T_x \simeq \mathbb{U} \wedge_{\{\pm 1\}} \bigwedge_{\tilde{x} \in \hat{\vec{A}}_x}
\bigl( \hat{\vec{A}}_{\sigma(\tilde{x})} \setminus \{\tilde{x}\} \bigr)
\quad \text{si } x \in A .
\]\[P_G = \prod_{x \in \hat{A}} T_x
= \mathbb{U}^{\hat{A}} \wedge_{\{\pm 1\}^{\hat{A}}}
\overbrace{\prod_{a \in \hat{A}} \varepsilon_a}^{S\varepsilon_G}\]
LaTeX source
\[
P_G = \prod_{x \in \hat{A}} T_x
= \mathbb{U}^{\hat{A}} \wedge_{\{\pm 1\}^{\hat{A}}}
\overbrace{\prod_{a \in \hat{A}} \varepsilon_a}^{S\varepsilon_G}
\]\[\begin{cases}
\varepsilon_a = \hat{\vec{A}}_s - \{a\} & \text{si } a \in I \\
\varepsilon_a = \bigwedge_{\tilde{a} \text{ sur } a} \bigl( \hat{\vec{A}}_{\sigma(\tilde{a})} \setminus \tilde{a} \bigr) & \text{si } a \in A = \hat{A} \setminus I
\end{cases}\]
LaTeX source
\[
\begin{cases}
\varepsilon_a = \hat{\vec{A}}_s - \{a\} & \text{si } a \in I \\
\varepsilon_a = \bigwedge_{\tilde{a} \text{ sur } a} \bigl( \hat{\vec{A}}_{\sigma(\tilde{a})} \setminus \tilde{a} \bigr) & \text{si } a \in A = \hat{A} \setminus I
\end{cases}
\]\[\mu^{\circ}_{\vec{A}} \times SP_G \longrightarrow SPM_{G^{\natural}}
\qquad \text{où } \mu^{\circ}_{\vec{A}} = (]0,1[)^{\vec{A}}\]
LaTeX source
\[
\mu^{\circ}_{\vec{A}} \times SP_G \longrightarrow SPM_{G^{\natural}}
\qquad \text{où } \mu^{\circ}_{\vec{A}} = (]0,1[)^{\vec{A}}
\]\[\pi_1(SP_G) = S\mathfrak{T}_G = \mathbb{Z}^{\hat{A}} \longrightarrow
\pi_1(SPM_{G^{\natural}}) = S\mathfrak{T}_{G^{\natural}}\]
LaTeX source
\[
\pi_1(SP_G) = S\mathfrak{T}_G = \mathbb{Z}^{\hat{A}} \longrightarrow
\pi_1(SPM_{G^{\natural}}) = S\mathfrak{T}_{G^{\natural}}
\]\[S\varepsilon_G \subset SP_G\]
LaTeX source
\[ S\varepsilon_G \subset SP_G \]
\[\mu^{\circ}_{\vec{A}} \times S\varepsilon_G \longrightarrow PM_{G^{\natural}}\]
LaTeX source
\[
\mu^{\circ}_{\vec{A}} \times S\varepsilon_G \longrightarrow PM_{G^{\natural}}
\]\[S\varepsilon_G \longrightarrow (S\mathfrak{T}_{G^{\natural}})\]
LaTeX source
\[
S\varepsilon_G \longrightarrow (S\mathfrak{T}_{G^{\natural}})
\]\[S_0\mathfrak{T}_G \longrightarrow S\mathfrak{T}_{G^{\natural}} .\]
LaTeX source
\[
S_0\mathfrak{T}_G \longrightarrow S\mathfrak{T}_{G^{\natural}} .
\]\[\alpha = (\alpha_a) \in \{\pm 1\}^{\hat{A}}\]
LaTeX source
\[
\alpha = (\alpha_a) \in \{\pm 1\}^{\hat{A}}
\]\[\xi \in \varepsilon_G\]
LaTeX source
\[ \xi \in \varepsilon_G \]
\[u_{\alpha,\xi} : \xi \longrightarrow \alpha.\xi = \xi'\]
LaTeX source
\[
u_{\alpha,\xi} : \xi \longrightarrow \alpha.\xi = \xi'
\]\[u_{\alpha,\xi}(t)_a =
\begin{cases}
\xi_a = \xi'_a & \text{si } \alpha_a = +1 \\
(\exp(\pi i t))\, \xi_a & \text{si } \alpha_a = -1
\end{cases}\]
LaTeX source
\[
u_{\alpha,\xi}(t)_a =
\begin{cases}
\xi_a = \xi'_a & \text{si } \alpha_a = +1 \\
(\exp(\pi i t))\, \xi_a & \text{si } \alpha_a = -1
\end{cases}
\]\[n(\alpha,\beta) \in \mathbb{Z}^{\hat{A}}\]
LaTeX source
\[
n(\alpha,\beta) \in \mathbb{Z}^{\hat{A}}
\]\[n(\alpha,\beta)_a =
\begin{cases}
0 & \text{si } \alpha_a = 1 \text{ ou } \beta_a = 1 \\
1 & \text{sinon (i.e. } \alpha_a = \beta_a = -1)
\end{cases}\]
LaTeX source
\[
n(\alpha,\beta)_a =
\begin{cases}
0 & \text{si } \alpha_a = 1 \text{ ou } \beta_a = 1 \\
1 & \text{sinon (i.e. } \alpha_a = \beta_a = -1)
\end{cases}
\]\[u_{a,\xi} : \xi \longrightarrow \sigma_a \xi\]
LaTeX source
\[
u_{a,\xi} : \xi \longrightarrow \sigma_a \xi
\]\[\varepsilon_G = \prod_{a \in A} \varepsilon_a \subset P_G = \prod_{a \in A} T_a\]
LaTeX source
\[
\varepsilon_G = \prod_{a \in A} \varepsilon_a \subset P_G = \prod_{a \in A} T_a
\]\[M_G \simeq M_{0,J}
\qquad \bigl( \simeq \mathbb{U}_{0,3} \simeq \mathbb{P}^1_{\mathbb{C}} \setminus \{0,1,\infty\} \bigr),\]
LaTeX source
\[
M_G \simeq M_{0,J}
\qquad \bigl( \simeq \mathbb{U}_{0,3} \simeq \mathbb{P}^1_{\mathbb{C}} \setminus \{0,1,\infty\} \bigr),
\]\[M_{0,J} = \text{\struck{$M_{0,4}$}}\; \mathrm{Rep}(J) \wedge_{\mathfrak{S}_4} M^{!}_{0,4}
= \mathrm{Rep}(J) \wedge_{\mathfrak{S}_4} \mathbb{U}_{0,3}\]
LaTeX source
\[
M_{0,J} = \text{\struck{$M_{0,4}$}}\; \mathrm{Rep}(J) \wedge_{\mathfrak{S}_4} M^{!}_{0,4}
= \mathrm{Rep}(J) \wedge_{\mathfrak{S}_4} \mathbb{U}_{0,3}
\]\[\mathfrak{S}_4 \longrightarrow \mathfrak{S}_3\]
LaTeX source
\[
\mathfrak{S}_4 \longrightarrow \mathfrak{S}_3
\]\[T_a = \mathbb{U} \wedge_{\{\pm 1\}} \varepsilon_a
\quad \text{pour} \quad
\varepsilon_a =
\begin{cases}
\hat{\vec{A}}_{\sigma(x)} \setminus x & \text{si } x \in I_s \\
\bigwedge_{\tilde{x} \in \hat{\vec{A}}_x} \bigl( \hat{\vec{A}}_{\sigma(\tilde{x})} \setminus \{\tilde{x}\} \bigr) & \text{si } x \in A
\end{cases}\]
LaTeX source
\[
T_a = \mathbb{U} \wedge_{\{\pm 1\}} \varepsilon_a
\quad \text{pour} \quad
\varepsilon_a =
\begin{cases}
\hat{\vec{A}}_{\sigma(x)} \setminus x & \text{si } x \in I_s \\
\bigwedge_{\tilde{x} \in \hat{\vec{A}}_x} \bigl( \hat{\vec{A}}_{\sigma(\tilde{x})} \setminus \{\tilde{x}\} \bigr) & \text{si } x \in A
\end{cases}
\]\[T_a = \bigwedge_{\tilde{a} \in A} T_{\tilde{a}}\]
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\[
T_a = \bigwedge_{\tilde{a} \in A} T_{\tilde{a}}
\]\[SP_X = \prod_{a \in \hat{A}} T_a\]
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\[
SP_X = \prod_{a \in \hat{A}} T_a
\]\[\varepsilon_{\hat{A}_1} = \prod_{a \in \hat{A}_1} \varepsilon_a ,\]
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\[
\varepsilon_{\hat{A}_1} = \prod_{a \in \hat{A}_1} \varepsilon_a ,
\]\[SP_X = \underbrace{\bigl( \mathbb{U}^{A_1} \wedge_{\{\pm 1\}^{A_1}} \varepsilon_{A_1} \bigr)}_{T_{A_1}} \times T^{Y}_{J} ,\]
LaTeX source
\[
SP_X = \underbrace{\bigl( \mathbb{U}^{A_1} \wedge_{\{\pm 1\}^{A_1}} \varepsilon_{A_1} \bigr)}_{T_{A_1}} \times T^{Y}_{J} ,
\]\[T^{Y}_{J} = \prod_{a \in \hat{A} \setminus \hat{A}_1} T_a\]
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\[
T^{Y}_{J} = \prod_{a \in \hat{A} \setminus \hat{A}_1} T_a
\]\[J = J_{\mathrm{I}} \amalg J_{\mathrm{II}} \amalg J_{\mathrm{III}}\]
LaTeX source
\[
J = J_{\mathrm{I}} \amalg J_{\mathrm{II}} \amalg J_{\mathrm{III}}
\]\[T^{Y}_{J} = T^{Y}_{J_{\mathrm{I}}} \times T^{Y}_{J_{\mathrm{II}}} \times T^{Y}_{J_{\mathrm{III}}}\]
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\[
T^{Y}_{J} = T^{Y}_{J_{\mathrm{I}}} \times T^{Y}_{J_{\mathrm{II}}} \times T^{Y}_{J_{\mathrm{III}}}
\]\[T^{Y}_{J_{\mathrm{I}}} = \prod_{i \in J_{\mathrm{I}}} T_i\]
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\[
T^{Y}_{J_{\mathrm{I}}} = \prod_{i \in J_{\mathrm{I}}} T_i
\]\[T^{Y}_{J_{\mathrm{II}}} =
\underbrace{\prod_{\alpha \in J_{\mathrm{II}}/\sigma}}_{0,\,1 \text{ ou } 2 \text{ facteurs}}
\;
\underbrace{\bigwedge_{\substack{i \in J_{\mathrm{II}} \\ \text{sur } \alpha}} T_i}_{2 \text{ facteurs}}\]
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\[
T^{Y}_{J_{\mathrm{II}}} =
\underbrace{\prod_{\alpha \in J_{\mathrm{II}}/\sigma}}_{0,\,1 \text{ ou } 2 \text{ facteurs}}
\;
\underbrace{\bigwedge_{\substack{i \in J_{\mathrm{II}} \\ \text{sur } \alpha}} T_i}_{2 \text{ facteurs}}
\]\[T^{Y}_{J_{\mathrm{III}}} = \prod_{i \in J_{\mathrm{III}}} T_i \wedge_{\{\pm 1\}} \tilde{J}_{\mathrm{III},i}\]
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\[
T^{Y}_{J_{\mathrm{III}}} = \prod_{i \in J_{\mathrm{III}}} T_i \wedge_{\{\pm 1\}} \tilde{J}_{\mathrm{III},i}
\]\[A_2 = A \setminus A_1 \simeq J_{\mathrm{I}} \amalg (J_{\mathrm{II}}/\sigma) \amalg J_{\mathrm{III}} .\]
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\[
A_2 = A \setminus A_1 \simeq J_{\mathrm{I}} \amalg (J_{\mathrm{II}}/\sigma) \amalg J_{\mathrm{III}} .
\]\[SP_G \simeq
\underbrace{T_{\hat{A}_1}}_{\text{$\mathbb{U}^{\hat{A}_1}$-torseur}}
\times
\underbrace{T_{\hat{A}_2}}_{\text{$\mathbb{U}^{A_2}$-torseur sur $M_{0,J}$}}\]
LaTeX source
\[
SP_G \simeq
\underbrace{T_{\hat{A}_1}}_{\text{$\mathbb{U}^{\hat{A}_1}$-torseur}}
\times
\underbrace{T_{\hat{A}_2}}_{\text{$\mathbb{U}^{A_2}$-torseur sur $M_{0,J}$}}
\]\[T_i = \mathbb{U} \times_{\{\pm1\}} \omega_Q\]
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\[
T_i = \mathbb{U} \times_{\{\pm1\}} \omega_Q
\]\[\text{\struck{$T^{Y}_{\hat{A}_2}$}} = \mathbb{U}^{A}\]
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\[
\text{\struck{$T^{Y}_{\hat{A}_2}$}} = \mathbb{U}^{A}
\]\[T^{Y}_{J_{\mathrm{I}}} = \mathbb{U}^{J_{\mathrm{I}}} \wedge_{\{\pm1\}} \omega_Q\]
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\[
T^{Y}_{J_{\mathrm{I}}} = \mathbb{U}^{J_{\mathrm{I}}} \wedge_{\{\pm1\}} \omega_Q
\]\[T^{Y}_{J_{\mathrm{II}}} \simeq \mathbb{U}^{J_2}\]
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\[
T^{Y}_{J_{\mathrm{II}}} \simeq \mathbb{U}^{J_2}
\]\[T^{Y}_{J_{\mathrm{III}}} \simeq \prod_{i\in J_{\mathrm{III}}} \mathbb{U}\wedge_{\{\pm1\}}
\underbrace{(\omega_Q \wedge \tilde{J}_{\mathrm{III}\,i})}_{\mathcal{E}_i}\]
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\[
T^{Y}_{J_{\mathrm{III}}} \simeq \prod_{i\in J_{\mathrm{III}}} \mathbb{U}\wedge_{\{\pm1\}}
\underbrace{(\omega_Q \wedge \tilde{J}_{\mathrm{III}\,i})}_{\mathcal{E}_i}
\]\[\mathcal{E}_{G,Q} = \mathcal{E}_{G,Y_Q}
= \underbrace{\mathcal{E}_{\hat{A}_1}}_{\prod_{a\in\hat{A}_1}\mathcal{E}_a}
\times \omega_Q \times
\underbrace{\mathcal{E}_{J_{\mathrm{III}}}}_{\prod_{i\in J_{\mathrm{III}}}\omega_Q\wedge\tilde{J}_{\mathrm{III}\,i}}\]
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\[
\mathcal{E}_{G,Q} = \mathcal{E}_{G,Y_Q}
= \underbrace{\mathcal{E}_{\hat{A}_1}}_{\prod_{a\in\hat{A}_1}\mathcal{E}_a}
\times \omega_Q \times
\underbrace{\mathcal{E}_{J_{\mathrm{III}}}}_{\prod_{i\in J_{\mathrm{III}}}\omega_Q\wedge\tilde{J}_{\mathrm{III}\,i}}
\]\[\gamma_{G_{\uncertain{1}}} = \{\pm1\}^{\hat{A}_1}\times\{\pm1\}\times\{\pm1\}^{J_{\mathrm{III}}}\]
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\[
\gamma_{G_{\uncertain{1}}} = \{\pm1\}^{\hat{A}_1}\times\{\pm1\}\times\{\pm1\}^{J_{\mathrm{III}}}
\]\[\gamma_G \longrightarrow \mathbb{U}^{\hat{A}} \simeq \mathbb{U}^{\hat{A}_1}\times\mathbb{U}^{J_{\mathrm{I}}}
\times\mathbb{U}^{J_{\mathrm{II}}/\sigma}\times\mathbb{U}^{J_{\mathrm{III}}}\]
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\[
\gamma_G \longrightarrow \mathbb{U}^{\hat{A}} \simeq \mathbb{U}^{\hat{A}_1}\times\mathbb{U}^{J_{\mathrm{I}}}
\times\mathbb{U}^{J_{\mathrm{II}}/\sigma}\times\mathbb{U}^{J_{\mathrm{III}}}
\]\[\gamma_G \longrightarrow \mathbb{U}^{\hat{A}_1}\times\check{\mathbb{U}}^{J_{\mathrm{I}}}\times\mathbb{U}^{J_{\mathrm{III}}}\]
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\[
\gamma_G \longrightarrow \mathbb{U}^{\hat{A}_1}\times\check{\mathbb{U}}^{J_{\mathrm{I}}}\times\mathbb{U}^{J_{\mathrm{III}}}
\]\[\{\pm1\}^{A_1} \hookrightarrow \mathbb{U}^{A_1}
\qquad\text{homom.\ évident déduit de } \{\pm1\}\hookrightarrow\mathbb{U}\]
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\[
\{\pm1\}^{A_1} \hookrightarrow \mathbb{U}^{A_1}
\qquad\text{homom.\ évident déduit de } \{\pm1\}\hookrightarrow\mathbb{U}
\]\[\{\pm1\} \longrightarrow \mathbb{U}^{J_{\mathrm{I}}},
\qquad
\{\pm1\}\xrightarrow{\ \mathrm{diag}\ }\{\pm1\}^{J_{\mathrm{I}}}
\ \text{\uncertain{$\Sigma$ t.\ d.\ car}}\]
LaTeX source
\[
\{\pm1\} \longrightarrow \mathbb{U}^{J_{\mathrm{I}}},
\qquad
\{\pm1\}\xrightarrow{\ \mathrm{diag}\ }\{\pm1\}^{J_{\mathrm{I}}}
\ \text{\uncertain{$\Sigma$ t.\ d.\ car}}
\]\[\{\pm1\}^{J_{\mathrm{III}}} \hookrightarrow \mathbb{U}^{J_{\mathrm{III}}}
\qquad\text{homom.\ évident.}\]
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\[
\{\pm1\}^{J_{\mathrm{III}}} \hookrightarrow \mathbb{U}^{J_{\mathrm{III}}}
\qquad\text{homom.\ évident.}
\]\[SP^{Y}_{G} \simeq \mathbb{U}^{\hat{A}} \wedge_{\gamma_G}
\mathcal{E}_{G,Q} \xrightarrow{\ \gamma_G\text{-tors.}\ }\]
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\[
SP^{Y}_{G} \simeq \mathbb{U}^{\hat{A}} \wedge_{\gamma_G}
\mathcal{E}_{G,Q} \xrightarrow{\ \gamma_G\text{-tors.}\ }
\]\[\mathcal{E}_{G,Q} = \mathcal{E}_{G,Y_Q} = \mathcal{E}_{\hat{A}_1}\times\mathcal{E}_{J_{\mathrm{III}}}\]
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\[
\mathcal{E}_{G,Q} = \mathcal{E}_{G,Y_Q} = \mathcal{E}_{\hat{A}_1}\times\mathcal{E}_{J_{\mathrm{III}}}
\]\[\gamma_G = \{\pm1\}^{\hat{A}_1}\times\{\pm1\}^{J_{\mathrm{III}}} = \{\pm1\}^{\hat{A}_1\amalg J_{\mathrm{III}}}\]
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\[
\gamma_G = \{\pm1\}^{\hat{A}_1}\times\{\pm1\}^{J_{\mathrm{III}}} = \{\pm1\}^{\hat{A}_1\amalg J_{\mathrm{III}}}
\]\[S_0T^{Y_Q}_{G} = S_0T^{Q}_{G}\]
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\[
S_0T^{Y_Q}_{G} = S_0T^{Q}_{G}
\]\[J_i \simeq \mathbb{U}\times_{\mu_3}\eta_i\]
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\[
J_i \simeq \mathbb{U}\times_{\mu_3}\eta_i
\]\[(1)\qquad (S,\hat{\vec{A}},\sigma,\bar{\sigma},g)\]
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\[
(1)\qquad (S,\hat{\vec{A}},\sigma,\bar{\sigma},g)
\]\[(2)\qquad
\underset{\text{arêtes}}{\hat{\vec{A}}}\xrightarrow{\ \sigma\ }\underset{\text{sommets}}{S}\xrightarrow{\ g\ }\mathbb{N}
\qquad \bar{\sigma}^2=\mathrm{id}_{\hat{\vec{A}}}\]
LaTeX source
\[
(2)\qquad
\underset{\text{arêtes}}{\hat{\vec{A}}}\xrightarrow{\ \sigma\ }\underset{\text{sommets}}{S}\xrightarrow{\ g\ }\mathbb{N}
\qquad \bar{\sigma}^2=\mathrm{id}_{\hat{\vec{A}}}
\]\[\begin{cases}
\hat{\vec{A}}^{\bar\sigma} = I \hookrightarrow \hat{A}\\[2pt]
\vec{A} = \hat{\vec{A}}\smallsetminus\hat{\vec{A}}^{\bar\sigma}\\[2pt]
A = \vec{A}/\bar\sigma \simeq \hat{A}\smallsetminus I
\end{cases}\]
LaTeX source
\[
\begin{cases}
\hat{\vec{A}}^{\bar\sigma} = I \hookrightarrow \hat{A}\\[2pt]
\vec{A} = \hat{\vec{A}}\smallsetminus\hat{\vec{A}}^{\bar\sigma}\\[2pt]
A = \vec{A}/\bar\sigma \simeq \hat{A}\smallsetminus I
\end{cases}
\]\[(3)\qquad \hat{\vec{A}}/\bar\sigma = \hat{A} \qquad (\text{arêtes})\]
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\[
(3)\qquad \hat{\vec{A}}/\bar\sigma = \hat{A} \qquad (\text{arêtes})
\]\[(4)\qquad \forall s\in S,\quad 2g_s+\hat{\nu}_s\geq 3
\quad\text{i.e.}\quad
\begin{cases}
g_s=0 \Rightarrow \hat{\nu}_s\geq 3\\
g_s=1 \Rightarrow \hat{\nu}_s\geq 1
\end{cases}\]
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\[
(4)\qquad \forall s\in S,\quad 2g_s+\hat{\nu}_s\geq 3
\quad\text{i.e.}\quad
\begin{cases}
g_s=0 \Rightarrow \hat{\nu}_s\geq 3\\
g_s=1 \Rightarrow \hat{\nu}_s\geq 1
\end{cases}
\]\[(5)\qquad \hat{\nu}_s = \nu_s + \vec{\nu}_s ,
\qquad \text{\struck{\ill{}}}\]
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\[
(5)\qquad \hat{\nu}_s = \nu_s + \vec{\nu}_s ,
\qquad \text{\struck{\ill{}}}
\]\[(6)\qquad \nu=\nu(G) = \sum_s \vec{\nu}_s = \operatorname{card}(\vec{A}^{\bar\sigma})
\quad(\text{\emph{ordre marqué} \struck{total} du graphe})\]
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\[
(6)\qquad \nu=\nu(G) = \sum_s \vec{\nu}_s = \operatorname{card}(\vec{A}^{\bar\sigma})
\quad(\text{\emph{ordre marqué} \struck{total} du graphe})
\]\[(7)\qquad c=c(G)=\tfrac12\sum_s\nu_s
\quad\text{nb des arêtes ordinaires : \emph{codim.\ modulaire} du graphe}\]
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\[
(7)\qquad c=c(G)=\tfrac12\sum_s\nu_s
\quad\text{nb des arêtes ordinaires : \emph{codim.\ modulaire} du graphe}
\]\[(8)\qquad g=\sum_{s\in S}g_s+\underbrace{h^1(G)}_{\text{connexité de }G}
\quad\text{\emph{genre virtuel} du graphe}\]
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\[
(8)\qquad g=\sum_{s\in S}g_s+\underbrace{h^1(G)}_{\text{connexité de }G}
\quad\text{\emph{genre virtuel} du graphe}
\]\[(9)\qquad \delta(G)=3g-3+\nu
\qquad(\text{\emph{dimension modulaire}, \add{\emph{\uncertain{composante}}}})\]
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\[
(9)\qquad \delta(G)=3g-3+\nu
\qquad(\text{\emph{dimension modulaire}, \add{\emph{\uncertain{composante}}}})
\]\[(10)\qquad \delta(G)+c(G)=\delta(\mathbf{G})\]
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\[
(10)\qquad \delta(G)+c(G)=\delta(\mathbf{G})
\]\[S\xrightarrow{\ u_s\ }S',
\qquad
\hat{\vec{A}}\xrightarrow{\ u_a\ }\hat{\vec{A}}{}'\amalg S'\]
LaTeX source
\[
S\xrightarrow{\ u_s\ }S',
\qquad
\hat{\vec{A}}\xrightarrow{\ u_a\ }\hat{\vec{A}}{}'\amalg S'
\]\[\vec{B}=u_a^{-1}(\hat{\vec{A}}{}'),\qquad
\vec{C}=\hat{\vec{A}}\smallsetminus\vec{B}=u_a^{-1}(S')\]
LaTeX source
\[
\vec{B}=u_a^{-1}(\hat{\vec{A}}{}'),\qquad
\vec{C}=\hat{\vec{A}}\smallsetminus\vec{B}=u_a^{-1}(S')
\]\[S\to S/R\to S'\]
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\[ S\to S/R\to S' \]
\[g(G_0(s'))=\sum_{s\in S(s')}g_s+h^1(G_0(s'))\]
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\[
g(G_0(s'))=\sum_{s\in S(s')}g_s+h^1(G_0(s'))
\]\[g_{s'}=g(G_0(s'))\]
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\[
g_{s'}=g(G_0(s'))
\]\[S(G_{0C})=S,\qquad
\vec{A}(G_{0C})=\vec{C}\quad(\text{image inv.\ de } C \text{ dans } \vec{A})\]
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\[
S(G_{0C})=S,\qquad
\vec{A}(G_{0C})=\vec{C}\quad(\text{image inv.\ de } C \text{ dans } \vec{A})
\]\[\left\{
\begin{aligned}
&S'=\pi_0(G_{0C})\\
&\hat{\vec{A}}{}'=\hat{\vec{A}}\smallsetminus\vec{C}\\
&\sigma':\hat{\vec{A}}{}'\to S' \text{ composé } \hat{\vec{A}}{}'\hookrightarrow\hat{\vec{A}}\xrightarrow{\ \sigma\ }S=S(G_{0C})\xrightarrow{\ \mathrm{can}\ }S'\\
&\bar\sigma' \text{ induit par } \bar\sigma\\
&g_{s'} = \text{genre virtuel de la comp.\ connexe } G_{0C,s'} \text{ du graphe pondéré } G_{0C}
\end{aligned}
\right.\]
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\[
\left\{
\begin{aligned}
&S'=\pi_0(G_{0C})\\
&\hat{\vec{A}}{}'=\hat{\vec{A}}\smallsetminus\vec{C}\\
&\sigma':\hat{\vec{A}}{}'\to S' \text{ composé } \hat{\vec{A}}{}'\hookrightarrow\hat{\vec{A}}\xrightarrow{\ \sigma\ }S=S(G_{0C})\xrightarrow{\ \mathrm{can}\ }S'\\
&\bar\sigma' \text{ induit par } \bar\sigma\\
&g_{s'} = \text{genre virtuel de la comp.\ connexe } G_{0C,s'} \text{ du graphe pondéré } G_{0C}
\end{aligned}
\right.
\]\[I\simeq I',\qquad \pi_0(G)\xrightarrow{\ \sim\ }\pi_0(G')\]
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\[
I\simeq I',\qquad \pi_0(G)\xrightarrow{\ \sim\ }\pi_0(G')
\]\[M_G\simeq M_{G'},\qquad \hat{M}_G\simeq\hat{M}_{G'},\qquad
\mathfrak{X}_{G'}\to\mathfrak{X}_G,\quad \hat{\mathfrak{X}}_{G'}\to\hat{\mathfrak{X}}_G\]
LaTeX source
\[
M_G\simeq M_{G'},\qquad \hat{M}_G\simeq\hat{M}_{G'},\qquad
\mathfrak{X}_{G'}\to\mathfrak{X}_G,\quad \hat{\mathfrak{X}}_{G'}\to\hat{\mathfrak{X}}_G
\]\[M_G\longrightarrow M_{G'},\qquad
\text{\struck{$\hat{M}_G\to\hat{M}_{G'}$}}\]
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\[
M_G\longrightarrow M_{G'},\qquad
\text{\struck{$\hat{M}_G\to\hat{M}_{G'}$}}
\]\[\mathfrak{X}_G\times_{M_{G'}}M_{G'}\longrightarrow\mathfrak{X}_G,
\qquad
\hat{\mathfrak{X}}_G\times_{\hat{M}_{G'}}\hat{M}_G\to\hat{\mathfrak{X}}_G\]
LaTeX source
\[
\mathfrak{X}_G\times_{M_{G'}}M_{G'}\longrightarrow\mathfrak{X}_G,
\qquad
\hat{\mathfrak{X}}_G\times_{\hat{M}_{G'}}\hat{M}_G\to\hat{\mathfrak{X}}_G
\]\[S'\xrightarrow{\ u_0\ }S,\qquad
\hat{\vec{A}}{}'=I'\amalg\vec{A}'\xrightarrow{\ u_1\ }\hat{\vec{A}}=I\amalg\vec{A}\]
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\[
S'\xrightarrow{\ u_0\ }S,\qquad
\hat{\vec{A}}{}'=I'\amalg\vec{A}'\xrightarrow{\ u_1\ }\hat{\vec{A}}=I\amalg\vec{A}
\]\[\tilde{X}=\coprod_{s\in S}\tilde{X}_s \longleftrightarrow \hat{\vec{A}}_k=\coprod_s(\hat{\vec{A}}_s)_k\]
LaTeX source
\[
\tilde{X}=\coprod_{s\in S}\tilde{X}_s \longleftrightarrow \hat{\vec{A}}_k=\coprod_s(\hat{\vec{A}}_s)_k
\]\[\tilde{X}'=\coprod_{s'\in S'}\tilde{X}'_{s'} \longrightarrow \hat{\vec{A}}{}'_k=\coprod_{s'\in S'}(\hat{\vec{A}}{}'_{s'})_k,
\qquad
\tilde{X}'_{s'}\overset{\mathrm{def}}{=}\tilde{X}_{u_0(s')}\]
LaTeX source
\[
\tilde{X}'=\coprod_{s'\in S'}\tilde{X}'_{s'} \longrightarrow \hat{\vec{A}}{}'_k=\coprod_{s'\in S'}(\hat{\vec{A}}{}'_{s'})_k,
\qquad
\tilde{X}'_{s'}\overset{\mathrm{def}}{=}\tilde{X}_{u_0(s')}
\]\[(\hat{\vec{A}}{}'_{s'})_k\overset{\mathrm{inj}}{\hookrightarrow}(\hat{\vec{A}}_s)_k\hookrightarrow\tilde{X}_s=\tilde{X}'_{s'}\]
LaTeX source
\[
(\hat{\vec{A}}{}'_{s'})_k\overset{\mathrm{inj}}{\hookrightarrow}(\hat{\vec{A}}_s)_k\hookrightarrow\tilde{X}_s=\tilde{X}'_{s'}
\]\[X'\longrightarrow X \qquad \text{\emph{morphisme fini net}}\]
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\[
X'\longrightarrow X \qquad \text{\emph{morphisme fini net}}
\]