Cote n° 80 · pages 2–69
· 93 displayed formulas · Systèmes de pseudo-droites : notes manuscrites (s.d.).
Inventory dating : [à partir de 1981]
Édition de démonstration
\[\mathrm{C}(I) \simeq \mathrm{C}_{\mathrm{aff}}(I_0) \longrightarrow \mathrm{C}_{n+1} \simeq \mathrm{C}(I)/\mathfrak{S}_I ,
\qquad
\mathrm{C}_{\mathrm{aff}}(n) \simeq \mathrm{C}_{\mathrm{aff}}(I_0)/\mathfrak{S}_{I_0} \simeq \mathrm{C}_{\mathrm{aff}}(I)/\mathfrak{S}_I\]
LaTeX source
\[
\mathrm{C}(I) \simeq \mathrm{C}_{\mathrm{aff}}(I_0) \longrightarrow \mathrm{C}_{n+1} \simeq \mathrm{C}(I)/\mathfrak{S}_I ,
\qquad
\mathrm{C}_{\mathrm{aff}}(n) \simeq \mathrm{C}_{\mathrm{aff}}(I_0)/\mathfrak{S}_{I_0} \simeq \mathrm{C}_{\mathrm{aff}}(I)/\mathfrak{S}_I
\]\[\mathrm{C}_{\mathrm{st}}(I) \simeq \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0)\]
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\[
\mathrm{C}_{\mathrm{st}}(I) \simeq \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0)
\]\[\widetilde{I}_0(\xi) \longrightarrow I_0(\xi)\]
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\[
\widetilde{I}_0(\xi) \longrightarrow I_0(\xi)
\]\[\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \longrightarrow \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0, \ldots)\]
LaTeX source
\[
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \longrightarrow \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0, \ldots)
\]\[\mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \overset{\mathrm{def}}{=} \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0, \pi_0) \subset \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0)\]
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\[
\mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \overset{\mathrm{def}}{=} \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0, \pi_0) \subset \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0)
\]\[\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \longrightarrow \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0)\]
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\[
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \longrightarrow \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0)
\]\[S \xrightarrow{\ \sim\ } \widetilde{I}_0(\xi)\]
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\[
S \xrightarrow{\ \sim\ } \widetilde{I}_0(\xi)
\]\[\alpha^{-1}(\xi^{q}) \xleftarrow{\ \sim\ } \mathfrak{S}_{I_0} \wedge^{\mathbb{D}_{I_0}} (\alpha^{!})^{-1}(\xi^{q})\]
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\[
\alpha^{-1}(\xi^{q}) \xleftarrow{\ \sim\ } \mathfrak{S}_{I_0} \wedge^{\mathbb{D}_{I_0}} (\alpha^{!})^{-1}(\xi^{q})
\]\[\operatorname{Card} \alpha^{-1}(\xi^{q}) = \Bigl( \frac{n!}{2n} \Bigr) \cdot \operatorname{Card} (\alpha^{!})^{-1}(\xi^{q}),
\qquad
\underbrace{\frac{n!}{2n}}_{} = \frac{(n-1)!}{2}\]
LaTeX source
\[
\operatorname{Card} \alpha^{-1}(\xi^{q}) = \Bigl( \frac{n!}{2n} \Bigr) \cdot \operatorname{Card} (\alpha^{!})^{-1}(\xi^{q}),
\qquad
\underbrace{\frac{n!}{2n}}_{} = \frac{(n-1)!}{2}
\]\[\operatorname{card} (\alpha^{!})^{-1}(\xi^{q}) = \frac{\operatorname{card} \mathbb{D}_{I_0}}{\operatorname{card} \bigl( G_{\xi} = \operatorname{Aut}(X, \Sigma, \Delta) \bigr)} = \frac{2n}{N(\xi)}\]
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\[
\operatorname{card} (\alpha^{!})^{-1}(\xi^{q}) = \frac{\operatorname{card} \mathbb{D}_{I_0}}{\operatorname{card} \bigl( G_{\xi} = \operatorname{Aut}(X, \Sigma, \Delta) \bigr)} = \frac{2n}{N(\xi)}
\]\[\operatorname{Card} \widetilde{\alpha}^{-1}(\xi^{q}) = \frac{4n}{N(\xi)}\]
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\[
\operatorname{Card} \widetilde{\alpha}^{-1}(\xi^{q}) = \frac{4n}{N(\xi)}
\]\[N(\xi) = \operatorname{card} \operatorname{Aut}(X, \Sigma, \Delta)\]
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\[
N(\xi) = \operatorname{card} \operatorname{Aut}(X, \Sigma, \Delta)
\]\[\xi^{q} = \mathrm{cl}_{\mathrm{aff}}(X, \Sigma, \Delta)\]
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\[
\xi^{q} = \mathrm{cl}_{\mathrm{aff}}(X, \Sigma, \Delta)
\]\[\operatorname{card} \widetilde{\mathrm{C}}_{\mathrm{st}}(S) = \frac{4n}{2} = 8\]
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\[
\operatorname{card} \widetilde{\mathrm{C}}_{\mathrm{st}}(S) = \frac{4n}{2} = 8
\]\[\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \overset{\mathrm{def}}{=}\]
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\[
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) \overset{\mathrm{def}}{=}
\]\[\bigl( \mathbb{D}_S \bigr)_{\widetilde{\xi}} \simeq \operatorname{Aut}_{\mathrm{ist}}(\mathbf{D}, \widetilde{\Sigma}) \simeq \operatorname{Aut}(X, \Sigma, \Delta)\]
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\[
\bigl( \mathbb{D}_S \bigr)_{\widetilde{\xi}} \simeq \operatorname{Aut}_{\mathrm{ist}}(\mathbf{D}, \widetilde{\Sigma}) \simeq \operatorname{Aut}(X, \Sigma, \Delta)
\]\[\widetilde{\mathrm{C}}_{\mathrm{st}}(S) / \mathbb{D}_S \simeq \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(n) \overset{\mathrm{def}}{=}\]
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\[
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) / \mathbb{D}_S \simeq \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(n) \overset{\mathrm{def}}{=}
\]\[\widetilde{\mathrm{C}}_{\mathrm{st}}(S) / \lbrace 1, a_S \rbrace \simeq \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \quad \bigl( \subset \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0) \bigr)\]
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\[
\widetilde{\mathrm{C}}_{\mathrm{st}}(S) / \lbrace 1, a_S \rbrace \simeq \mathrm{C}^{!}_{\mathrm{aff}\,\mathrm{st}}(I_0) \quad \bigl( \subset \mathrm{C}_{\mathrm{aff}\,\mathrm{st}}(I_0) \bigr)
\]\[\exists !\ X' \longrightarrow \widetilde{Z}\]
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\[
\exists !\ X' \longrightarrow \widetilde{Z}
\]\[48 = 2 \cdot 4! \qquad \frac{5!}{2 \cdot 4!} = \frac{5}{2}\]
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\[
48 = 2 \cdot 4! \qquad \frac{5!}{2 \cdot 4!} = \frac{5}{2}
\]\[3 \times 2 \times 3 \qquad 12 = \frac{8 \times 3 \times 2}{4}\]
LaTeX source
\[
3 \times 2 \times 3 \qquad 12 = \frac{8 \times 3 \times 2}{4}
\]\[a + a' \leqslant 2 , \qquad b + b' \geqslant 2 ,
\qquad 2 - a' < b \quad \text{i.e.} \quad a' + b > 2\]
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\[
a + a' \leqslant 2 , \qquad b + b' \geqslant 2 ,
\qquad 2 - a' < b \quad \text{i.e.} \quad a' + b > 2
\]\[\varphi_H : \mathbf{S} \xrightarrow{\ \varphi_H\ } D_H , \qquad \mathbf{S} = V^{*} / \mathbf{R}^{*}_{+}\]
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\[
\varphi_H : \mathbf{S} \xrightarrow{\ \varphi_H\ } D_H , \qquad \mathbf{S} = V^{*} / \mathbf{R}^{*}_{+}
\]\[\varphi : \mathbf{S} \longrightarrow \prod D_{H_i} = \mathbf{E}\]
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\[
\varphi : \mathbf{S} \longrightarrow \prod D_{H_i} = \mathbf{E}
\]\[\text{Soit } \Sigma = \varphi(\mathbf{S}) \subset \mathbf{E}\]
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\[
\text{Soit } \Sigma = \varphi(\mathbf{S}) \subset \mathbf{E}
\]\[Z^{*}_{-\alpha} = - Z^{*}_{\alpha} \qquad \text{pour } \alpha \in \Sigma\]
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\[
Z^{*}_{-\alpha} = - Z^{*}_{\alpha} \qquad \text{pour } \alpha \in \Sigma
\]\[\overline{V'_{H_i}} = V'_{H_i} \cup H_i , \qquad \overline{V''_{H_i}} = V''_{H_i} \cup H_i ,\]
LaTeX source
\[
\overline{V'_{H_i}} = V'_{H_i} \cup H_i , \qquad \overline{V''_{H_i}} = V''_{H_i} \cup H_i ,
\]\[\overline{Z}^{*}_{\alpha} = \bigcup_{\substack{\beta \in \Sigma \\ \text{tel que } i \in I \\ \text{implique } k_i(\beta) \leqslant k_i(\alpha)}} Z^{*}_{\beta}
= \bigcup_{\substack{\beta \in \Sigma \\ \beta \preceq \alpha}} Z^{*}_{\beta}\]
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\[
\overline{Z}^{*}_{\alpha} = \bigcup_{\substack{\beta \in \Sigma \\ \text{tel que } i \in I \\ \text{implique } k_i(\beta) \leqslant k_i(\alpha)}} Z^{*}_{\beta}
= \bigcup_{\substack{\beta \in \Sigma \\ \beta \preceq \alpha}} Z^{*}_{\beta}
\]\[k_i : \mathbf{E} \longrightarrow \mathbb{D}_{H_i} \qquad (i \in I)\]
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\[
k_i : \mathbf{E} \longrightarrow \mathbb{D}_{H_i} \qquad (i \in I)
\]\[\beta \leqslant \alpha \iff \beta \in \overline{\lbrace \alpha \rbrace} , \quad \text{et} \quad
\overline{Z}^{*}_{\alpha} = Z^{*}_{\overline{\lbrace \alpha \rbrace}}\]
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\[
\beta \leqslant \alpha \iff \beta \in \overline{\lbrace \alpha \rbrace} , \quad \text{et} \quad
\overline{Z}^{*}_{\alpha} = Z^{*}_{\overline{\lbrace \alpha \rbrace}}
\]\[Z^{*}_{A} = \bigcup_{\alpha \in A} Z^{*}_{\alpha} .\]
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\[
Z^{*}_{A} = \bigcup_{\alpha \in A} Z^{*}_{\alpha} .
\]\[\overline{Z^{*}_{A}} = Z^{*}_{\overline{A}}\]
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\[
\overline{Z^{*}_{A}} = Z^{*}_{\overline{A}}
\]\[\Bigl( \bigcup_{\alpha \in A} \overline{Z}_{\alpha} = \bigcup_{\alpha \in A} Z_{\overline{\lbrace \alpha \rbrace}} = Z_{\bigcup_{\alpha \in A} \overline{\lbrace \alpha \rbrace}} , \quad \bigcup_{\alpha \in A} \overline{\lbrace \alpha \rbrace} = \overline{A} \Bigr)\]
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\[
\Bigl( \bigcup_{\alpha \in A} \overline{Z}_{\alpha} = \bigcup_{\alpha \in A} Z_{\overline{\lbrace \alpha \rbrace}} = Z_{\bigcup_{\alpha \in A} \overline{\lbrace \alpha \rbrace}} , \quad \bigcup_{\alpha \in A} \overline{\lbrace \alpha \rbrace} = \overline{A} \Bigr)
\]\[\boxed{\mathrm{I}'_{3,2},\ \mathrm{II}_5,\ \mathrm{III}_6}\]
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\[
\boxed{\mathrm{I}'_{3,2},\ \mathrm{II}_5,\ \mathrm{III}_6}
\]\[(1) \qquad
\begin{array}{cl}
\Phi = \coprod_{i \in I} \Phi_i & \operatorname{card} 6 \\
\downarrow & \\
I & \operatorname{card} 3
\end{array}\]
LaTeX source
\[
(1) \qquad
\begin{array}{cl}
\Phi = \coprod_{i \in I} \Phi_i & \operatorname{card} 6 \\
\downarrow & \\
I & \operatorname{card} 3
\end{array}
\]\[(2) \qquad
\begin{array}{l}
C_2 \simeq \Phi \simeq \mathfrak{P}_1(\Phi) \\
C_1 \simeq \lbrace A \in \mathfrak{P}_2(\Phi) \mid A \to I \text{ injectif} \rbrace \\
C_0 \simeq \lbrace A \in \mathfrak{P}_3(\Phi) \mid A \to I \text{ injectif (donc bij.)} \rbrace
\end{array}\]
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\[
(2) \qquad
\begin{array}{l}
C_2 \simeq \Phi \simeq \mathfrak{P}_1(\Phi) \\
C_1 \simeq \lbrace A \in \mathfrak{P}_2(\Phi) \mid A \to I \text{ injectif} \rbrace \\
C_0 \simeq \lbrace A \in \mathfrak{P}_3(\Phi) \mid A \to I \text{ injectif (donc bij.)} \rbrace
\end{array}
\]\[(3) \qquad
\left\lbrace
\begin{array}{l}
O_0 = C_2 \simeq \Phi \simeq \mathfrak{P}_1(\Phi) \\
O_1 = C_1 \simeq \lbrace A \in \mathfrak{P}_2(\Phi) \mid A \hookrightarrow I \rbrace \\
O_2 = C_0 \simeq \lbrace A \in \mathfrak{P}_3(\Phi) \mid A \hookrightarrow I \rbrace
\end{array}
\right.\]
LaTeX source
\[
(3) \qquad
\left\lbrace
\begin{array}{l}
O_0 = C_2 \simeq \Phi \simeq \mathfrak{P}_1(\Phi) \\
O_1 = C_1 \simeq \lbrace A \in \mathfrak{P}_2(\Phi) \mid A \hookrightarrow I \rbrace \\
O_2 = C_0 \simeq \lbrace A \in \mathfrak{P}_3(\Phi) \mid A \hookrightarrow I \rbrace
\end{array}
\right.
\]\[(4) \qquad \widetilde{C}_i = C_i/\lbrace 1, \underline{a} \rbrace
\qquad \widetilde{O}_i = O_i/\lbrace 1, \underline{a} \rbrace\]
LaTeX source
\[
(4) \qquad \widetilde{C}_i = C_i/\lbrace 1, \underline{a} \rbrace
\qquad \widetilde{O}_i = O_i/\lbrace 1, \underline{a} \rbrace
\]\[(5) \qquad
\begin{array}{ccc}
\text{cubes comb.\ orientés} & \xrightarrow{\ \approx\ } & \text{cubes comb.\ gauches} \\
\wr\downarrow & & \wr\downarrow \\
\text{octaèdres comb.\ orientés} & \xrightarrow{\ \approx\ } & \text{octaèdres comb.\ gauches}
\end{array}\]
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\[
(5) \qquad
\begin{array}{ccc}
\text{cubes comb.\ orientés} & \xrightarrow{\ \approx\ } & \text{cubes comb.\ gauches} \\
\wr\downarrow & & \wr\downarrow \\
\text{octaèdres comb.\ orientés} & \xrightarrow{\ \approx\ } & \text{octaèdres comb.\ gauches}
\end{array}
\]\[(6) \qquad \operatorname{or}(C) \simeq \operatorname{or}(O) \simeq \Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or}(I)\]
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\[
(6) \qquad \operatorname{or}(C) \simeq \operatorname{or}(O) \simeq \Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or}(I)
\]\[(7) \qquad A_i = \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} \Phi_\alpha
\qquad \text{(\uncertain{ens.}\ des deux arêtes de $\widetilde{O}$ portées par $D_i$)}\]
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\[
(7) \qquad A_i = \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} \Phi_\alpha
\qquad \text{(\uncertain{ens.}\ des deux arêtes de $\widetilde{O}$ portées par $D_i$)}
\]\[(8) \qquad S_i \simeq I \setminus \lbrace i \rbrace
\qquad \text{(ens.\ des \add{deux} sommets sur la droite)}\]
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\[
(8) \qquad S_i \simeq I \setminus \lbrace i \rbrace
\qquad \text{(ens.\ des \add{deux} sommets sur la droite)}
\]\[(9) \qquad \Omega_i \simeq S_i \wedge A_i \simeq (I \setminus \lbrace i \rbrace) \wedge \underbrace{\bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} \Phi_\alpha}_{\Phi_i \wedge \operatorname{or}(I) \text{ à cause de (6 bis)}}\]
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\[
(9) \qquad \Omega_i \simeq S_i \wedge A_i \simeq (I \setminus \lbrace i \rbrace) \wedge \underbrace{\bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} \Phi_\alpha}_{\Phi_i \wedge \operatorname{or}(I) \text{ à cause de (6 bis)}}
\]\[(10) \qquad \underbrace{\bigwedge_{i \in I} A_i}_{\bigwedge_{\substack{\alpha, i \in I \\ \alpha \neq i}} \Phi_\alpha \;=\; \bigwedge_{\alpha \in I} \Phi_\alpha^2 \;\simeq\; \underline{1}} \simeq \underline{1}\]
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\[
(10) \qquad \underbrace{\bigwedge_{i \in I} A_i}_{\bigwedge_{\substack{\alpha, i \in I \\ \alpha \neq i}} \Phi_\alpha \;=\; \bigwedge_{\alpha \in I} \Phi_\alpha^2 \;\simeq\; \underline{1}} \simeq \underline{1}
\]\[(11) \qquad \bigwedge_{i \in I} \Omega_i \overset{\varphi}{\simeq} \bigwedge_{i \in I} S_i
\quad \Bigl[\simeq \bigwedge_{i \in I} (I \setminus \lbrace i \rbrace) \simeq \operatorname{or}(I)\Bigr]\]
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\[
(11) \qquad \bigwedge_{i \in I} \Omega_i \overset{\varphi}{\simeq} \bigwedge_{i \in I} S_i
\quad \Bigl[\simeq \bigwedge_{i \in I} (I \setminus \lbrace i \rbrace) \simeq \operatorname{or}(I)\Bigr]
\]\[(11\ \text{bis}) \qquad \bigwedge_{i \in I} \Omega_i \simeq \operatorname{or}(I)\]
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\[
(11\ \text{bis}) \qquad \bigwedge_{i \in I} \Omega_i \simeq \operatorname{or}(I)
\]\[(12) \qquad \Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or}(I) \simeq \underline{1}\]
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\[
(12) \qquad \Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or}(I) \simeq \underline{1}
\]\[(13) \qquad
\left\lbrace
\begin{array}{l}
\Omega_i \simeq \Phi_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace) \\[1ex]
\Phi_i \simeq \Omega_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace)
\end{array}
\right.\]
LaTeX source
\[
(13) \qquad
\left\lbrace
\begin{array}{l}
\Omega_i \simeq \Phi_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace) \\[1ex]
\Phi_i \simeq \Omega_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace)
\end{array}
\right.
\]\[(14) \qquad
\left\lbrace
\begin{array}{l}
\bigwedge_{i \in I} \Omega_i \simeq \bigwedge_{i \in I} \Phi_i \\[1ex]
\underbrace{\Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Omega_i)_{i \in I}}
\simeq
\underbrace{\Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Phi_i)_{i \in I}}
\end{array}
\right.\]
LaTeX source
\[
(14) \qquad
\left\lbrace
\begin{array}{l}
\bigwedge_{i \in I} \Omega_i \simeq \bigwedge_{i \in I} \Phi_i \\[1ex]
\underbrace{\Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Omega_i)_{i \in I}}
\simeq
\underbrace{\Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Phi_i)_{i \in I}}
\end{array}
\right.
\]\[S_i \simeq I - \lbrace i \rbrace\]
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\[ S_i \simeq I - \lbrace i \rbrace \]
\[\Omega_i = S_i \wedge A_i = (I - \lbrace i \rbrace) \wedge A_i\]
LaTeX source
\[ \Omega_i = S_i \wedge A_i = (I - \lbrace i \rbrace) \wedge A_i \]
\[\begin{aligned}
\Phi_i &= \Omega_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace) \\
&\simeq A_i \wedge \bigwedge_{\alpha \in I} (I \setminus \lbrace \alpha \rbrace)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\Phi_i &= \Omega_i \wedge \bigwedge_{\alpha \in I \setminus \lbrace i \rbrace} (I \setminus \lbrace \alpha \rbrace) \\
&\simeq A_i \wedge \bigwedge_{\alpha \in I} (I \setminus \lbrace \alpha \rbrace)
\end{aligned}
\]\[(15) \qquad \bigwedge_{i \in I} \Omega_i \simeq \bigwedge_{i \in I} \Phi_i \simeq \Bigl(\bigwedge_{i \in I} A_i\Bigr) \wedge \underbrace{\bigwedge_{\alpha \in I} (I \setminus \lbrace \alpha \rbrace)}_{\operatorname{or} I}\]
LaTeX source
\[
(15) \qquad \bigwedge_{i \in I} \Omega_i \simeq \bigwedge_{i \in I} \Phi_i \simeq \Bigl(\bigwedge_{i \in I} A_i\Bigr) \wedge \underbrace{\bigwedge_{\alpha \in I} (I \setminus \lbrace \alpha \rbrace)}_{\operatorname{or} I}
\]\[(15\ \text{bis}) \qquad
\underbrace{\Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or} I}_{\substack{\text{or.\ de l'oct.} \\ \text{défini par } (\Omega_i)_{i \in I}}}
\simeq
\underbrace{\Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or}(I)}_{\substack{\text{or.\ de l'oct.} \\ \text{défini par } (\Phi_i)_{i \in I}}}
\simeq \bigwedge_{i \in I} A_i\]
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\[
(15\ \text{bis}) \qquad
\underbrace{\Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or} I}_{\substack{\text{or.\ de l'oct.} \\ \text{défini par } (\Omega_i)_{i \in I}}}
\simeq
\underbrace{\Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or}(I)}_{\substack{\text{or.\ de l'oct.} \\ \text{défini par } (\Phi_i)_{i \in I}}}
\simeq \bigwedge_{i \in I} A_i
\]\[(16) \qquad (\text{oct.\ gauches}) \longrightarrow \overbrace{\text{graphes octaédraux gauches}}^{\text{gr.\ oct.\ gauches}^{+}}\]
LaTeX source
\[
(16) \qquad (\text{oct.\ gauches}) \longrightarrow \overbrace{\text{graphes octaédraux gauches}}^{\text{gr.\ oct.\ gauches}^{+}}
\]\[\text{oct} \longrightarrow \text{gr.\ oct.} \longrightarrow \text{gr.\ oct g.}\]
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\[
\text{oct} \longrightarrow \text{gr.\ oct.} \longrightarrow \text{gr.\ oct g.}
\]\[\text{oct} \longrightarrow \text{oct.\ gauches} \qquad (\text{surjectif sur les isom.}).\]
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\[
\text{oct} \longrightarrow \text{oct.\ gauches} \qquad (\text{surjectif sur les isom.}).
\]\[\text{oct.\ g} \longrightarrow \text{gr.\ oct.\ gauches},\]
LaTeX source
\[
\text{oct.\ g} \longrightarrow \text{gr.\ oct.\ gauches},
\]\[\omega = (\omega_I)_{\substack{I \in \mathfrak{P}_3(J) \\ \bigcap_{i \in I} D_i = \emptyset}}\]
LaTeX source
\[
\omega = (\omega_I)_{\substack{I \in \mathfrak{P}_3(J) \\ \bigcap_{i \in I} D_i = \emptyset}}
\]\[(18) \qquad \mathcal{X} \longmapsto (K, (D_i)_{i \in J}, \omega)\]
LaTeX source
\[
(18) \qquad \mathcal{X} \longmapsto (K, (D_i)_{i \in J}, \omega)
\]\[\operatorname{Isom}(\mathcal{X}, \mathcal{X}') \xrightarrow{\ \sim\ } \operatorname{Isom}\bigl((K, (D_i)_{i \in J}, \omega), (K', (D'_{i'})_{i' \in J'}, \omega')\bigr)\]
LaTeX source
\[
\operatorname{Isom}(\mathcal{X}, \mathcal{X}') \xrightarrow{\ \sim\ } \operatorname{Isom}\bigl((K, (D_i)_{i \in J}, \omega), (K', (D'_{i'})_{i' \in J'}, \omega')\bigr)
\]\[(19) \qquad \vec{\mathfrak{Z}}_s \subset \mathcal{C}_{\vec{A}_s}\]
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\[
(19) \qquad \vec{\mathfrak{Z}}_s \subset \mathcal{C}_{\vec{A}_s}
\]\[(20) \qquad \psi_{\vec{a}} : \vec{\mathfrak{Z}}_s \xrightarrow{\ \sim\ } \vec{\mathfrak{Z}}_{s'}\]
LaTeX source
\[
(20) \qquad \psi_{\vec{a}} : \vec{\mathfrak{Z}}_s \xrightarrow{\ \sim\ } \vec{\mathfrak{Z}}_{s'}
\]\[(21) \qquad \psi_{\overleftarrow{a}} = (\psi_{\vec{a}})^{-1} \text{)}\]
LaTeX source
\[
(21) \qquad \psi_{\overleftarrow{a}} = (\psi_{\vec{a}})^{-1} \text{)}
\]\[(22) \qquad \text{si } \rho \in \vec{\mathfrak{Z}}_s \text{, et si } \nu_s \text{ est le nb de } D_i \text{ passant par } s \text{, on a } \rho^{\nu_s}(\vec{D}_i) = (-\vec{D}_i)\]
LaTeX source
\[
(22) \qquad \text{si } \rho \in \vec{\mathfrak{Z}}_s \text{, et si } \nu_s \text{ est le nb de } D_i \text{ passant par } s \text{, on a } \rho^{\nu_s}(\vec{D}_i) = (-\vec{D}_i)
\]\[(23) \qquad \underbrace{c_0 - c_1}_{1 - c(K)} + c_2 = 1 \qquad \text{soit} \qquad c_2 = c(K)\]
LaTeX source
\[
(23) \qquad \underbrace{c_0 - c_1}_{1 - c(K)} + c_2 = 1 \qquad \text{soit} \qquad c_2 = c(K)
\]\[(24) \qquad \mathfrak{Z}^{1}_s \subset \mathcal{C}_{I_s}\]
LaTeX source
\[
(24) \qquad \mathfrak{Z}^{1}_s \subset \mathcal{C}_{I_s}
\]\[(25) \qquad I_s \xrightarrow{\ \varphi = \varphi_{s,\Delta}\ } \text{ens.\ } S_\Delta \text{ des sommets de } K \text{ sur } \Delta\]
LaTeX source
\[
(25) \qquad I_s \xrightarrow{\ \varphi = \varphi_{s,\Delta}\ } \text{ens.\ } S_\Delta \text{ des sommets de } K \text{ sur } \Delta
\]\[D_i \cap \Delta = \lbrace \varphi(D_i) \rbrace\]
LaTeX source
\[ D_i \cap \Delta = \lbrace \varphi(D_i) \rbrace \]
\[\varphi_{s,\Delta} : I_s \hookrightarrow S_\Delta\]
LaTeX source
\[
\varphi_{s,\Delta} : I_s \hookrightarrow S_\Delta
\]\[(A1) \qquad \mathfrak{Z}^{1}_{s,\Delta} \text{ est indépendant du choix de } \Delta \in I \setminus I_s \text{, soit } \mathfrak{Z}^{1}_s .\]
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\[
(A1) \qquad \mathfrak{Z}^{1}_{s,\Delta} \text{ est indépendant du choix de } \Delta \in I \setminus I_s \text{, soit } \mathfrak{Z}^{1}_s .
\]\[(26) \qquad \vec{I}_s \longrightarrow I_s \qquad \text{de degré } 2\]
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\[
(26) \qquad \vec{I}_s \longrightarrow I_s \qquad \text{de degré } 2
\]\[(27) \qquad
\begin{cases}
\text{si } \vec{u} \in \vec{\mathfrak{Z}}_s \ (\subset \mathcal{C}_{\vec{I}_s}) \text{, } u \text{ par passage au quotient dans (26),} \\
\text{on a } u \in \mathcal{C}_{I_s} \text{, et on a } u \in \mathfrak{Z}^{1}_s .
\end{cases}\]
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\[
(27) \qquad
\begin{cases}
\text{si } \vec{u} \in \vec{\mathfrak{Z}}_s \ (\subset \mathcal{C}_{\vec{I}_s}) \text{, } u \text{ par passage au quotient dans (26),} \\
\text{on a } u \in \mathcal{C}_{I_s} \text{, et on a } u \in \mathfrak{Z}^{1}_s .
\end{cases}
\]\[(28) \qquad \omega_{i_0} \xrightarrow[\sim]{\ \alpha_{i_0}\ } \omega_{i_1} \xrightarrow[\sim]{\ \alpha_{i_1}\ } \omega_{i_2} \ \cdots\ \xrightarrow[\sim]{\ \alpha_{i_{\nu_s - 2}}\ } \omega_{i_{\nu_s - 1}}
\qquad (\nu = \nu_s)\]
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\[
(28) \qquad \omega_{i_0} \xrightarrow[\sim]{\ \alpha_{i_0}\ } \omega_{i_1} \xrightarrow[\sim]{\ \alpha_{i_1}\ } \omega_{i_2} \ \cdots\ \xrightarrow[\sim]{\ \alpha_{i_{\nu_s - 2}}\ } \omega_{i_{\nu_s - 1}}
\qquad (\nu = \nu_s)
\]\[(29)\]
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\[ (29) \]
\[(30) \qquad \vec{u}_{s,-\vec{\Delta}} = \vec{u}_{s,\vec{\Delta}}^{\,-1} , \qquad
\vec{\mathfrak{Z}}_{s,\Delta} = \lbrace \vec{u}_{s,\vec{\Delta}}, \vec{u}_{s,-\vec{\Delta}} \rbrace \subset \mathcal{C}_{\vec{I}_s}\]
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\[
(30) \qquad \vec{u}_{s,-\vec{\Delta}} = \vec{u}_{s,\vec{\Delta}}^{\,-1} , \qquad
\vec{\mathfrak{Z}}_{s,\Delta} = \lbrace \vec{u}_{s,\vec{\Delta}}, \vec{u}_{s,-\vec{\Delta}} \rbrace \subset \mathcal{C}_{\vec{I}_s}
\]\[\vec{\mathfrak{Z}}_{s,\Delta'} = \vec{\mathfrak{Z}}_{s,\Delta} \qquad (\text{soit } \vec{\mathfrak{Z}}_s)\]
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\[
\vec{\mathfrak{Z}}_{s,\Delta'} = \vec{\mathfrak{Z}}_{s,\Delta} \qquad (\text{soit } \vec{\mathfrak{Z}}_s)
\]\[(31) \qquad \vec{I} \setminus \vec{I}_s \longrightarrow \vec{\mathfrak{Z}}_s\]
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\[
(31) \qquad \vec{I} \setminus \vec{I}_s \longrightarrow \vec{\mathfrak{Z}}_s
\]\[\psi_{\vec{a}} : \vec{\mathfrak{Z}}_s \longrightarrow \vec{\mathfrak{Z}}_{s'}\]
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\[
\psi_{\vec{a}} : \vec{\mathfrak{Z}}_s \longrightarrow \vec{\mathfrak{Z}}_{s'}
\]\[\vec{J} = \vec{I} \setminus (\vec{I}_s \cup \vec{I}_{s'})\]
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\[
\vec{J} = \vec{I} \setminus (\vec{I}_s \cup \vec{I}_{s'})
\]\[(32)\]
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\[ (32) \]
\[(33)\]
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\[ (33) \]
\[(34) \qquad \vec{\psi}_{\vec{a}} = \psi_{\vec{b}'} \, \psi_{\vec{b}}\]
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\[
(34) \qquad \vec{\psi}_{\vec{a}} = \psi_{\vec{b}'} \, \psi_{\vec{b}}
\]\[\begin{pmatrix} \text{arrangements simples} \\ \text{de } 4 \text{ ps.\ droites} \end{pmatrix}
\longrightarrow
\begin{pmatrix} \text{arr.\ } 1\text{-dimensionnels} \\ \text{simples de } 4 \\ \text{ps-droites} \end{pmatrix}\]
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\[
\begin{pmatrix} \text{arrangements simples} \\ \text{de } 4 \text{ ps.\ droites} \end{pmatrix}
\longrightarrow
\begin{pmatrix} \text{arr.\ } 1\text{-dimensionnels} \\ \text{simples de } 4 \\ \text{ps-droites} \end{pmatrix}
\]\[\begin{array}{r}
5 \\
15 = 5 + 10 \\
10 = 5 + 5 \\
1 \\
\hline
31
\end{array}\]
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\[
\begin{array}{r}
5 \\
15 = 5 + 10 \\
10 = 5 + 5 \\
1 \\
\hline
31
\end{array}
\]\[\Omega(D) \xrightarrow{\ \sim\ } \Omega(D') \qquad \text{(via } \Omega_s(X)\text{)} \ ?\]
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\[
\Omega(D) \xrightarrow{\ \sim\ } \Omega(D') \qquad \text{(via } \Omega_s(X)\text{)} \ ?
\]\[\Omega(D) \simeq \Omega(D')\]
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\[ \Omega(D) \simeq \Omega(D') \]
\[\Omega(\Gamma) \simeq \Omega(D') , \qquad \Omega(\Gamma') \simeq \Omega(D)\]
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\[ \Omega(\Gamma) \simeq \Omega(D') , \qquad \Omega(\Gamma') \simeq \Omega(D) \]
\[\Omega(\Gamma) \simeq \Omega(\Gamma')\]
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\[ \Omega(\Gamma) \simeq \Omega(\Gamma') \]
\[\varphi_{s;D,D'} : \Omega(D) \simeq \Omega(D')\]
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\[
\varphi_{s;D,D'} : \Omega(D) \simeq \Omega(D')
\]\[\varphi_{s;D,D'} = \varphi_{s';D,D'}\]
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\[
\varphi_{s;D,D'} = \varphi_{s';D,D'}
\]\[\rho_{\vec{D}}(s u_1 u'_1) = (s u_2 u'_2) .\]
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\[
\rho_{\vec{D}}(s u_1 u'_1) = (s u_2 u'_2) .
\]\[(\rho_{\vec{D},s})^{n}(\vec{\Delta}) = -\vec{\Delta} .\]
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\[
(\rho_{\vec{D},s})^{n}(\vec{\Delta}) = -\vec{\Delta} .
\]