\[
SRT(X) =
\]

\[
S_{*}T^{!}(X) \qquad \mathbb{U}^{I} \qquad \mathfrak{S}_{I}
\qquad\qquad I = \pi_0(\partial X)
\]

\[
\mathbf{Z}^{I} \qquad T(X) \qquad \mathbb{U}^{I} \qquad \mathfrak{S}_{I}
\]

\[
SRT^{!}(X) = \mathbf{R}^{I} \cdot S_{*}T^{!}(X)
\]

\[
\mathbf{Z}^{I} \hookrightarrow S_{*}T^{!} \subset SRT^{!} \subset SRT
\qquad \text{i.e.\ on a}
\]

\[
SRT/\mathbf{Z}^{I} \overset{\mathrm{def}}{=} SUT(X)
\quad\big\rbrace\ \mathfrak{S}_I
\]

\[
(*)\qquad 1 \to \mathbb{U}^{I} \to SUT(X) \to T(X) \to 1
\]

\[
\underline{s} = (s_i) \in \prod_{i \in I = \pi_0(\partial X)} T_i
\]

\[
S_{*}T^{!}(X, \underline{s}) \hookrightarrow SRT(X)
\]

\[
S_{*}T^{!}(X, \underline{s}) \wedge^{\mathbf{Z}^{I}} \mathbf{R}^{I}
\simeq SRT(X)
\]

\[
\varphi_{\underline{s}}(T(X)) \subset SUT(X)
\]

\[
SRT(X) \to SUT(X)
\]

\[
\varphi_{\underline{s}'} = \mathrm{int}(u) \circ \varphi_{\underline{s}}
\]

\[
S_{*}T(X, \underline{s}') = \mathrm{int}(\tilde{u})\, S_{*}T(X, \underline{s}).
\]

\[
T_i(n_i) \subset T_i,
\]

\[
\underline{s} \in \prod_{i\in I} T_i^{\wedge n_i},
\]

\[
\underline{s}^{0} \in \prod T_i \quad\text{et}\quad
S_{*}T(X, \underline{s}^{0}) \subset SRT(X),
\]

\[
S_{*}T(X, \underline{s}) = S_{*}T(X, \underline{s}^{0}) \cdot
\prod_{i\in I} \tfrac{1}{n_i}\mathbf{Z}_i
\qquad\qquad \mathbf{Z}^{I} \subset SRT(X).
\]

\[
\mathbf{Z}/4\mathbf{Z} \simeq \mu_4(\mathbf{C}) \simeq
\]

\[
\mathbb{E}\mathrm{l}_0 \overset{\mathrm{def}}{\simeq} \mathbf{C}/\mathbf{Z}[i]
\]

\[
\mathbb{E}\mathrm{l}_0(S) \simeq \mathbb{E}\mathrm{l}_0 \wedge^{\mathbf{Z}/4\mathbf{Z}} S
\]

\[
\boxed{\mathbb{E}\mathrm{l}_0(S) =: X_S = V_S/\Gamma_S}\ .
\]

\[
\mu_4(\mathbf{C}) \simeq \mathrm{Aut}(X_S).
\]

\[
(X_S)^{i\text{-fix}} = \lbrace x \in X_S \mid ix = x \rbrace
= \lbrace e, \eta_S \rbrace .
\]

\[
\struck{\mathbb{E}\mathrm{l}_0(S)}\ X_S/\pm 1
\]

\[
\mathbf{Z}/6\mathbf{Z} \simeq \mu_6(\mathbf{C}) \simeq
\]

\[
\mathbf{Z}[\zeta] = \mathbf{Z}[T]/(1 + T + T^{2}) \qquad
(\zeta = \exp 2i\pi/3)
\]

\[
\mathbb{E}\mathrm{l}_1 \simeq \mathbf{C}/\mathbf{Z}[\zeta]
\]

\[
\mathbb{E}\mathrm{l}_1(S) \simeq \mathbb{E}\mathrm{l}_1
\wedge^{\mathbf{Z}/6\mathbf{Z}} S
\]

\[
X_S = \mathbb{E}\mathrm{l}_1(S) \simeq V_S/\Gamma_S
\]

\[
\text{à} \quad S/\pm 1,
\]

\[
s_0 = 4 + 6 + \struck{4}\,3 = 22 \qquad\struck{\ill{}}
\]

\[
s_1 = 12 + 4.\struck{9}\ \ldots = \struck{4}\,8\ \ldots
\qquad
s_2 = 4.7 = 28
\]

\[
I = S_{Q} = \coprod_{d \in D_Q} d \qquad
A_{Q} = \prod_{d \in D_Q} d \qquad
C_{Q} = \bigwedge_{d \in D_Q} d
\]

\[
\omega_{Q} = C_{Q} \wedge D_{Q} \qquad
\omega_{I} \simeq \omega_{J} \simeq C_{Q} \simeq J \setminus \lbrace Q \rbrace
\]

\[
\begin{array}{l}
\quad a\ b\ c\ d \\
(a\,d)(b\,c) = Q_0 \\
(b\,d)(a\,c) = Q_1 \\
(c\,d)(a\,b) = Q_{\infty}
\end{array}
\]

\[
(\text{tétraèdre \emph{orienté} } + \text{diagonale})
\longleftrightarrow (\text{carré} + \text{codiagonale})
\]

\[
0 \to V(J) \to \mathfrak{S}_{I} \to \mathfrak{S}_{J} \to 1
\qquad\qquad
\boxed{\begin{cases} \rho^{3} = \sigma^{2} = 1 \\ \rho\sigma = \ell \end{cases}}
\]

\[
\ill{}\ :\qquad \mu_6 \times (\mathbf{Z}/2\mathbf{Z})^{2} , \qquad
\mu_4 \times (\mathbf{Z}/2\mathbf{Z})^{2}
\]

\[
\mathrm{diag}_{Q} \simeq \omega_{Q} \qquad
\mathrm{diag}_{Q} \wedge \omega_{Q} \simeq \mathrm{codiag}_{Q}
\]

\[
\begin{array}{llll}
A & \text{triangle} & \Sigma_{A} & (0,3) \\
\vec{T} & \text{tétraèdre orienté} & \Sigma_{T} & (0,4) \\
\vec{Q} & \text{carré} & \Sigma_{Q} & \\
\vec{\mathbb{Q}} & \text{carré orienté} & E_{\vec{Q}} & (1,1) \\
\vec{H} & \text{hexagone orienté} & E_{\vec{H}} & \\
 & & & (0,5) \\
 & & & (1,2)
\end{array}
\]

\[
\sigma_0^{2} = \sigma_1^{2} = \sigma_2^{2} = (\sigma_0\sigma_2)^{2}
= (\sigma_0\sigma_1)^{3} = \underline{1}
\]

\[
\begin{array}{l}
4 \text{ hexagones} \\
16 \text{ arêtes} \\
10 \text{ sommets} \\
\text{(dont 4 ramifiés d'ordre \ldots)}
\end{array}
\Longrightarrow
\begin{array}{l}
8 \text{ hexagones} \\
32 \text{ arêtes} \\
16 \text{ sommets}
\end{array}
\]

\[
\begin{array}{l}
2 \text{ hexag.} \\
9 \text{ arêtes} \\
6 \text{ sommets}
\end{array}
\]

\[
\begin{array}{l}
6 \text{ sommets (dont 4 ramifiés)} \\
8 \text{ arêtes} \\
4\ \text{carrés}
\end{array}
\Longrightarrow
\begin{array}{l}
8 \text{ sommets d'ordre 4} \\
16 \text{ arêtes} \\
8\ \text{carrés}
\end{array}
\]

\[
G \simeq \mathbb{D}_5 , \qquad \widetilde{G} \simeq \mathbb{D}_5 \times
\lbrace 1, \sigma \rbrace .
\]

\[
G \simeq G_J \simeq \mathfrak{S}_3 , \qquad \widetilde{G} \simeq
\mathfrak{S}_3 \times \lbrace 1, \sigma \rbrace .
\]

\[
I = \coprod_{\alpha \in \delta} \alpha \quad (\delta = \text{ens.\ des
diagonales } \alpha), \qquad \omega \in J \wedge
\bigwedge_{\alpha \in \delta} \alpha , \qquad G = \widetilde{G} \simeq
\mathbf{Z}/4\mathbf{Z} .
\]

\[
G \simeq \mathbf{Z}/2\mathbf{Z} , \qquad \widetilde{G} \simeq
\mathbf{Z}/2\mathbf{Z} \times \lbrace 1, \sigma \rbrace .
\]

\[
G \simeq \mathbf{Z}/2\mathbf{Z} , \qquad \widetilde{G} \simeq
\mathbf{Z}/2\mathbf{Z} \times \lbrace 1, \sigma \rbrace .
\]

\[
G = \lbrace 1 \rbrace , \qquad \widetilde{G} = \lbrace 1, \sigma \rbrace
\]

\[
\tau^{0}_{r} : \mathrm{II}_1(J, E) \longrightarrow
\mathrm{II}_1(u_{\omega}(r)) \quad \text{où } u_{\omega} = \omega
\times (1) \text{ sur } J \times E \bigr]
\]

\[
G = \mathbf{Z}/2\mathbf{Z} , \qquad \widetilde{G} = \mathbf{Z}/2\mathbf{Z}
\times \lbrace 1, \sigma \rbrace .
\]

\[
G = 1 , \qquad \widetilde{G} = \lbrace 1, \sigma \rbrace .
\]

\[
G = \mathbf{Z}/2\mathbf{Z} , \qquad \widetilde{G} \simeq
\mathbf{Z}/2\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z} \simeq
\mathbf{Z}/2\mathbf{Z} \times \lbrace 1, \sigma \rbrace .
\]

\[
\begin{array}{llll}
\text{niveau 2} & \mathrm{I}_{10}, \mathrm{I}_{6}, \mathrm{I}_{4}
  & 12+20+30 = 62 \\
\text{niveau 1} & \mathrm{II}_{2}, \mathrm{II}'_{2}, \mathrm{II}_{1}
  & 60+60+120 = 240 \\
\text{niveau 0} & \mathrm{III}_{2}, \mathrm{III}'_{1},
  \mathrm{III}'_{2} & 60+120+60 = 240 \\
 & & \text{total } \overline{542}
\end{array}
\]

\[
\begin{array}{rll}
180 & \left\lbrace \begin{array}{l}
  60 = 3\times 20 = 2\times 30 \\
  120 = 4\times 30 = 2\times 60
  \end{array}\right. &
  \begin{array}{l} f_{B,s} \\ f_{\vec{Q},\alpha} \end{array} \\[1ex]
420 & \left\lbrace \begin{array}{l}
  60 = 1\times 60 = 1\times 60 \\
  120 = 1\times 120 = 1\times 120 \\
  60 = 1\times 60 = 5\times 12 \\
  60 = 1\times 60 = 3\times 20 \\
  120 = 1\times 120 = 6\times 20
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,s} \\ \tau^{0}_{r} \\ \gamma_{\pi,s} \\
  \gamma'_{\pi,s} \\ \gamma^{0}_{r} \end{array} \\[1ex]
1020 & \left\lbrace \begin{array}{l}
  120 = 2\times 60 = 1\times 120 \\
  120 = 2\times 60 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  3\times 120 = 3\times 40 = 3\times 40 \\
  120 = 2\times 60 = 2\times 60 \\
  60 = 1\times 60 = 2\times 30
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,\rho} \\ \gamma_{\pi,\rho} \\ \tau'_{r} \\
  \gamma^{1}_{r} \\ \gamma^{1\,\prime}_{r} \\ \gamma^{2}_{r} \\
  \gamma^{2\,\prime}_{r} \end{array}
\end{array}
\]

\[
\overline{\overline{1620}} = 5\times 60 + 11\times 120
\]

\[
K = K^{g} \amalg \text{orbites d'ordre } \nu, \quad \text{donc}\quad
2 \leqslant \nu \leqslant 5 .
\]

\[
K = \lbrace 0, 1, \infty, \lambda, \bar{\lambda} \rbrace , \quad
\lambda \in \text{demi-plan de Poincaré},
\]

\[
\text{dimension modulaire} \qquad 2 \qquad 1 \qquad 0
\]

\[
\begin{array}{llll}
\text{nb comp.} & 1 & 10 & 15 \\
\text{nb réalisations spéciales} & 12 & 10\times 3\times 2 = 60 &
15\times 2\times 2 = 60 \\
 & & + 10\times 2\times 6 = 120 & \\
\end{array}
\]

\[
\lambda = \frac{(\xi + \eta)^{2}}{\xi \wedge \eta} = \frac{\xi}{\eta} +
\frac{\eta}{\xi} + 2
\]

\[
\lambda' = \lambda - 2 = \frac{\xi}{\eta} + \frac{\eta}{\xi} \neq 2
\]

\[
z^{2} - \mu z + 1 = 0 ,
\]

\[
z^{4} - \mu z^{2} + 1 = 0
\]

\[
I = \Bigl\lbrace \pm\sqrt{\tfrac{1}{2}\bigl(\mu \pm \sqrt{\mu^{2} -
4}\bigr)} \Bigr\rbrace = \Bigl\lbrace \pm\sqrt{\mu' \pm
\sqrt{\mu'^{2} - 1}} \Bigr\rbrace
\]

\[
e = \xi \wedge \eta = \eta \wedge \xi \in T_4 = T^{\wedge 4} \simeq
L_4^{*} \quad (\text{où } L_4 = L_1^{\otimes 4})
\]

\[
3g_s - 3 + \nu_s \geqslant 0 \qquad
2g_s - 2 + \nu_s \geqslant 1 \qquad
2g_s + \nu_s \geqslant 3
\]

\[
\begin{array}{l|l}
\text{cercles de découpage} & \text{graphe pondéré} \\
\text{cercles de raccord} & \text{graphe bipondéré} \\
 & \text{graphe pondéré marqué}
\end{array}
\]

\[
\boxed{S_{*}\mathcal{T}^{\pm 1}_{g,\nu} \longrightarrow
\mathcal{T}^{\pm 1}_{g,\nu} \longrightarrow \mathrm{Ens}(\nu) \times
[\pm 1]}
\]

\[
\mathcal{T}^{+}_{\mathrm{top}} \quad\text{ou}\quad \mathcal{T}_{\mathrm{top}}
\]

\[
\mathcal{T}^{\pm}_{\mathrm{top}}
\]

\[
\mathcal{T}^{\pm}_{\mathrm{top}} \xrightarrow{\ \chi\ } [\pm 1]
\]

\[
S_{*}\mathcal{T}D^{+}_{\mathrm{top}} \quad\text{ou}\quad S_{*}\mathcal{T}D_{\mathrm{top}},
\]

\[
S \xrightarrow{\ \gamma\ } \mathbf{N}
\]

\[
\vec{A}^{\sigma} = A_{\ell} \xrightarrow{\ r\ } \mathbf{N}
\]

\begin{align*}
\widetilde{S} &= S \sqcup A_{\ell}, \\
\widetilde{A} &= \vec{A}/\sigma = A \sqcup A_{\ell}, \\
\vec{A} &= \vec{A}_0 \sqcup \vec{A}_{\ell},
  \qquad \vec{A}_0 = \vec{A} \setminus \vec{A}^{\sigma},
  \quad \vec{A}_{\ell} = \vec{A}^{\sigma},
\end{align*}

\[
\sigma(s) = \sigma_{\ell}(s) + \sigma_0(s),
\]

\[
\bigl(S,\ (\vec{A} \setminus \vec{A}^{\sigma}),\
\sigma\,|\,(\vec{A} \setminus \vec{A}^{\sigma}),\
o\,|\,(\vec{A} \setminus \vec{A}^{\sigma})\bigr),
\]

\[
A_{\ell} \longrightarrow S .
\]

\[
(*)\qquad \Gamma : S_{*}\mathcal{T}D^{\pm}_{\mathrm{top}} \longrightarrow
\text{(Gra bipond)}
\]

\[
X \ \text{connexe} \iff \Gamma_{\mathrm{MD}}(X, \widetilde{\Sigma}, R)\ \text{connexe}
\]

\[
\pi_0(X) \simeq \pi_0(\Gamma_{\mathrm{MD}}(X, \widetilde{\Sigma}, R)).
\]

\[
g = \gamma(s), \qquad \nu = \operatorname{card} A .
\]

\[
\struck{A\,(\widetilde{R}, \alpha,\ \rho : R \to \vec{A})}\qquad (R_a)_{a \in A}
\]

\[
\omega \mapsto u_{\omega} : \omega(a) \longrightarrow \mathrm{Circ}(R_a)
\]

\[
r : A \to \mathbf{N}, \qquad a \mapsto \operatorname{card} \rho^{-1}(a)
\]

\[
(R_a)_{a \in A},
\]

\[
R_a = R \cap \widetilde{\Sigma}_a
\]

\[
\vec{R}
\]

\[
D_{\infty} = \lbrace \sigma_0, \sigma_1 \mid \sigma_0^2 = \sigma_1^2 = 1 \rbrace
\]

\[
\vec{R} \longrightarrow \vec{A}
\]

\[
D_{\infty} \xrightarrow{\ \varepsilon\ } \lbrace \pm 1 \rbrace
\]

\[
\vec{R}^{+}_{\ell} \subset \vec{R}\,|\,\vec{A}_{\ell}
\]

\[
S_{*}\mathcal{T}D^{\pm}_{\mathrm{top}} \longrightarrow (\mathrm{Gr.pondrep})
\]

\[
S_{*}\mathcal{T}D^{\pm}_{\mathrm{top}} \longrightarrow (\mathrm{Gr.pondrep.})
\]

\[
S\mathcal{T}D^{\pm}_{\mathrm{top}} \longrightarrow (\mathrm{Gr.pond.rep}),
\]

\[
\Sigma^{+} \setminus R \cap \Sigma^{+}
\]

\[
B^{+} \xrightarrow{\ r'\ } \mathbf{N},
\]

\[
R_a \quad (a \in A)
\]

\[
\mathbf{Z}_a/(r(a)) \qquad \bigl(\mathbf{Z}_a = \mathbf{Z} \wedge_{\lbrace \pm 1 \rbrace} \omega(a)\bigr),
\]

\[
A' \subset A^{+},
\]

\[
R_a \ \text{et}\ R_{a'} \qquad (r = r(a) = r(a')),
\]

\[
R_a \wedge_{\mathbf{Z}/r\mathbf{Z}} R_{a'} .
\]

\[
J \xrightarrow{\ \pi\ } \mathbb{N}
\]

\[
\alpha_s + \beta_s = \alpha_t + \beta_t
\]

\[
\begin{array}{lll}
\beta + \gamma = \alpha' & \qquad & \alpha = \dfrac{\beta' + \gamma' - \alpha'}{2} = \dfrac{\alpha' + \beta' + \gamma'}{2} - \alpha' \\[2ex]
\gamma + \alpha = \beta' & & \beta = \dfrac{\gamma' + \alpha' - \beta'}{2} \\[2ex]
\alpha + \beta = \gamma' & & \gamma = \dfrac{\alpha' + \beta' - \gamma'}{2}
\end{array}
\]

\[
\pi_0(\mathcal{T}_{\mathrm{top}}) \xrightarrow{\ \sim\ } \pi_0(\mathcal{T}^{\pm}_{\mathrm{top}}) \simeq
\]

\[
S\mathcal{T}^{\pm}_{\mathrm{top}} \xrightarrow{\ \varepsilon\ } [\pm 1]
\]