\[
  H^{*}(P, F) \longrightarrow H^{*}(\mathring{P}, F) \quad \text{un isom}
  \qquad (\text{si } F \text{ loc. constant})
\]

\[
  H^{*}_{\partial P}(P, F) = 0
\]

\[
  \underline{H}^{*}_{\partial P}(F) = 0 .
\]

\[
  \cdots \longrightarrow H^{i}_{!}(\mathring{P}, F) \longrightarrow H^{i}(P, F)
  \longrightarrow H^{i}(\partial P, F) \longrightarrow \cdots
\]

\[
  H^{i}_{!}(\mathring{P}, F) \simeq H^{n-i}(\mathring{P}, \check{F} \otimes T)^\vee
  \simeq H^{n-i}(P, \check{F} \otimes T)^\vee
\]

\[
  \varphi : H^{n-i}(P, \check{F} \otimes T)^\vee \longrightarrow H^{i}(P, F)
\]

\[
  \varphi \in H^{i}(P, F) \otimes H^{n-i}(P, \check{F} \otimes T)
\]

\[
  \widetilde{\partial P} \simeq \partial P \times D^{1} .
\]

\[
  \begin{array}{ccccccc}
    \to & H^{i}_{!}(P - \widetilde{\partial P}) & \to & H^{i}(P) & \to & H^{i}(\widetilde{\partial P}) & \to \cdots \\
        & \downarrow & & \| & & \downarrow{\wr} & \\
    \to & H^{i}_{!}(P - \partial P) & \to & H^{i}(P) & \to & H^{i}(\partial P) & \to \cdots
  \end{array}
\]

\[
  H^{i}_{!}(P - \widetilde{\partial P}) \simeq H^{i}(P - \partial P)
\]

\[
  \begin{array}{ccccccc}
    \to & H^{i}_{P'}(P, F) & \Longrightarrow & H^{i}(P, F) & \to & H^{i}(P - P', F) & \to \cdots \\
        & \downarrow & & \| & & \updownarrow{\scriptstyle 2} & \\
    \to & H^{i}(P, \partial P; F) & \longrightarrow & H^{i}(P, F) & \to & H^{i}(\partial P, F) & \to
  \end{array}
  \qquad
  \begin{array}{c} (P, P - P') \\ \uparrow \\ (P, \partial P) \end{array}
\]

\[
  H^{i}_{P'}(P, F) \simeq H^{i}(P \bmod \partial P, F) \simeq H^{i}_{!}(\mathring{P}, F)
\]

\[
  H^{i}_{P'}(P, F) \xrightarrow{\ \sim\ } H^{i}_{!}(\mathring{P}' \bmod \partial P', F)
  \simeq H^{i}_{!}(\mathring{P}', F)
\]

\[
  (P', \partial P') \to (P, P - P') .
\]

\[
  W = P \cup Q, \qquad P \cap Q = \partial P = \partial Q .
\]

\[
  W - R = \mathring{P} \amalg \mathring{Q} .
\]

\[
  \cdots \to H^{i}(W, F) \to H^{i}(W - R, F) \to H^{i+1}(R, F \otimes \mathcal{T}_{R})
  \to H^{i+1}(W, F)
\]

\[
  H^{i}(W - R) = H^{i}(\mathring{P}) \times H^{i}(\mathring{Q}) = H^{i}(P) \times H^{i}(Q)
\]

\[
  \cdots \to H^{i}_{!}(W - T, F) \to H^{i}_{!}(W, F) \to H^{i}_{!}(R, F) \to \cdots
\]

\[
  \begin{aligned}
  H^{i}_{!}(W - T, F) &\simeq H^{i}_{!}(\mathring{P}, F) \times H^{i}_{!}(\mathring{Q}, F) \\
  &\simeq H^{n-i}(\mathring{P}, \check{F} \otimes \mathcal{T}_W)^\vee \times
     H^{n-i}(\mathring{Q}, \check{F} \otimes \mathcal{T}_W)^\vee \\
  &\simeq H^{n-i}(P, \check{F} \otimes \mathcal{T}_W)^\vee \times
     H^{n-i}(Q, \check{F} \otimes \mathcal{T}_W)^\vee
  \end{aligned}
\]

\[
  \cdots \to H^{i}_{P}(W, F) \to H^{i}(W, F) \to H^{i}(W - P, F) \to \cdots
\]

\[
  H^{i}(W - P, F) \simeq H^{i}(\mathring{Q}, F) \xleftarrow{\ \sim\ } H^{i}(Q, F)
\]

\[
  H^{i}_{P}(\widetilde{P}, F) \xrightarrow{\ \sim\ }
  H^{i}_{!}(\widetilde{P} \bmod \partial\widetilde{P}, F) \simeq H^{i}_{!}(\mathring{P}, F),
  \qquad
  H^{i}_{P}(\widetilde{P}, F) \simeq H^{i}_{P}(W, F) \ \ (\text{excision})
\]

\[
  \cdots \to H^{i}_{!}(\mathring{P}, F) \to H^{i}(W, F) \to H^{i}(Q, F) \to \cdots
\]

\[
  H^{i}_{!}(\mathring{P}, F) \simeq H^{n-i}(P, \check{F} \otimes \mathcal{T}_W)^\vee
\]

\[
  \boxed{\begin{array}{ccc}
    S_{!}(W, R) & \text{dual de} & S(W, R) \\
    \uparrow & & \downarrow \\
    S(W, P) \simeq S_{!}(W, Q) & \text{dual de} & S(W, Q) \simeq S_{!}(W, P) \\
    \uparrow & & \downarrow \\
    S(W, R) \simeq MV(W; P, Q) & \text{dual de} & S_{!}(W, R) \simeq MV_{!}(W, R)
  \end{array}}
\]

\[
  \begin{gathered}
  H^{i}_{Q}(W) \to H^{i}(W) \to H^{i}(W - Q) \simeq H^{i}(P) \\
  H^{i}_{Q}(W) \simeq H^{i}_{!}(W - P) = H^{i}_{!}(Q - \partial Q)
  \simeq \bigl(H^{n-i}(Q - \partial Q)\bigr)^\vee = H^{n-i}(Q)^\vee \\
  H^{i}_{Q}(W) \times H^{n-i}(Q)
  \end{gathered}
\]

\[
  \boxed{\begin{array}{c|l}
    \begin{array}{ccc} & \Psi & \\ \swarrow & & \searrow \\ \Phi & & \Phi' \\ \searrow & & \swarrow \\ & \Psi' & \end{array}
    &
    \begin{array}{l}
      \Psi : H(P), H(Q), H(T), H(W) \\
      \Psi' : \ldots\ \text{dual} \\
      \Phi : H(P)^\vee, H(Q), H(W) \\
      \Phi' : H(P), H(Q)^\vee, H(W)
    \end{array}
  \end{array}}
\]

\[
  \left.\begin{aligned}
    \Phi &\simeq S(W, P) \simeq S_{!}(W, Q) \\
    \Phi' &\simeq S(W, Q) \simeq S_{!}(W, P)
  \end{aligned}\right\} (\Phi, \Phi' \text{ duals})
  \qquad
  \left.\begin{aligned}
    \Psi &\simeq S(W, T) \simeq MV(W; P, Q) \\
    \Psi' &\simeq S_{!}(W, T) \simeq MV_{!}(W; P, Q)
  \end{aligned}\right\} (\Psi, \Psi' \text{ duals})
\]

\[
  P \cap Q = \partial P = \partial Q, \qquad P \cup Q = W
\]

\[
  \begin{array}{ccc}
    P \longrightarrow S & \qquad & Q \longrightarrow T \\
    \wr & & \uparrow \\
    M & & N
  \end{array}
\]

\[
  \begin{array}{ccc}
    \partial P \longrightarrow S & \qquad & \partial Q \longrightarrow T \\
    \wr & & \uparrow \\
    \partial M & & \partial N
  \end{array}
\]

\[
  \alpha : S \simeq \partial N \qquad \beta : T \simeq \partial M
\]

\[
  \sigma : \partial P \simeq S \times \partial M \qquad
  \tau : \partial Q \simeq T \times \partial N
\]

\[
  (\alpha, \sigma) : \partial P \simeq \partial N \times \partial M, \qquad
  (\beta, \tau) : \partial Q \simeq \partial M \times \partial N
\]

\[
  \varphi : \partial P \simeq \partial Q .
\]

\[
  W - Q = P - \partial P \hookrightarrow P
\]

\[
  W - P = Q - \partial Q \hookrightarrow Q .
\]

\[
  \begin{array}{cccc}
    W & P & Q & R \\
    V' & X'_{\eta} = V_{\eta} & V'_{s} & V'_{s} \times \eta
  \end{array}
  \qquad\qquad
  \begin{array}{ccc} & W & \\ P & & Q \\ & R & \end{array}
\]

\[
  \begin{array}{cccc}
    M & N & T = \dot{M}, & S = \dot{N} \\
    X'_{\eta} & e & V'_{s} & \eta \sim S^{1}
  \end{array}
\]

\[
  H^{i}(Q) \times H^{n-i-1}(P) \longrightarrow H^{n-1}(R) \simeq \Lambda
\]

\[
  H^{n-i}(P)^\vee \longrightarrow H^{i}(P)
\]

\[
  \binom{5}{2} = \frac{5.4.3}{12.} = 10
\]

\[
  \begin{array}{ccc}
    123 & 234 & 345 \\
    124 & 235 & \\
    125 & 245 & \\
    134 & & \\
    135 & & \\
    145 & &
  \end{array}
  \qquad 10 + 10 + 2
\]

\[
  \begin{array}{ccc}
    \emptyset & T \sim S & R \\
    5 + 5 & 2 + 2 & 2 \\
    3 & 3 + 3 & \\
    2 + 2 & 1 + 1 & \\
    1 + 2 & 1 + 1 & \\ \hline
    \boxed{20} & \boxed{14} & \boxed{2}
  \end{array}
  \qquad
  \begin{array}{c}
    M \sim N \\
    2 + 2 \\ \hline
    \boxed{4}
  \end{array}
\]

\[
  L^{\bullet\pi} = \mathbb{R}\Gamma(W_0 \times S^{1}) \simeq
  \mathbb{R}\Gamma(W_0) \overset{\mathbf{L}}{\otimes}_{\Lambda} \mathbb{R}\Gamma(S^{1})
  \simeq \mathbb{R}\Gamma(W_0) \times \mathbb{R}\Gamma(W_0)(-1)
\]

\[
  \begin{aligned}
    \mathbb{R}\Gamma(P) \simeq \mathbb{R}\Gamma(W - W_0) &\xrightarrow[\ \alpha\ ]{\text{compatible avec cup}} \mathbb{R}\Gamma(W_0) \\
    \mathbb{R}\Gamma(P) \simeq \mathbb{R}\Gamma(W - W_0) &\xrightarrow{\ \beta\ } \mathbb{R}\Gamma(W_0)(-1)
  \end{aligned}
\]

\[
  u_1 = \rho_0, \qquad u_2 = \mathrm{id}_1
\]

\[
  \mathbb{R}(\Gamma, \pi)_{!}(\mathring{M}) \simeq
  \mathbb{R}\mathrm{Hom}_{\Lambda}(K^\bullet, \Lambda)[2n],
  \qquad K^\bullet = \mathbb{R}(\Gamma, \pi)(M),
\]

\[
  \boxed{\ u : \mathbb{R}\mathrm{Hom}_{\Lambda}(K^\bullet, \Lambda)[-2n] \longrightarrow K^\bullet\ }
  \qquad \text{dans } D(\Lambda[\pi]),
\]

\[
  \xi \in H^{2n}\bigl(K^\bullet \overset{\mathbf{L}}{\otimes}_{\Lambda} K^\bullet\bigr).
\]

\[
  \boxed{\ u_1 : \mathbb{R}\Gamma(W - W_0)^\vee[-2n-1] \longrightarrow \mathbb{R}\Gamma(W - W_0)\ }
\]

\[
  \boxed{\begin{aligned}
    \lambda &: \mathbb{R}\Gamma(W_0) \times \mathbb{R}\Gamma(W - W_0) \longrightarrow \Lambda[-2n] \\
    \mu &: \mathbb{R}\Gamma(W_0) \times \mathbb{R}\Gamma(W - W_0) \longrightarrow \Lambda[-(2n-1)]
  \end{aligned}}
\]

\[
  \begin{aligned}
    \lambda &: H^{2n-i}(W_0) \times H^{i}(W - W_0) \longrightarrow \Lambda \\
    \mu &: H^{2n-1-i}(W_0) \times H^{i}(W - W_0) \longrightarrow \Lambda
  \end{aligned}
\]

\[
  \begin{aligned}
    \lambda^i &: H^{i}(W - W_0) \longrightarrow H^{i-1}(W_0) && \text{je dis que c'est } \beta \\
    \mu^i &: H^{i}(W - W_0) \longrightarrow H^{i}(W_0) && \text{je dis que c'est } \alpha
  \end{aligned}
\]

\[
  L^{\bullet\,\mathrm{triv}} \times K^\bullet \longrightarrow \Lambda[-2n+1]
\]

\[
  \boxed{\ \text{accoupl\supplied{ement}}\quad \mathbb{R}\Gamma(W_0)^{\mathrm{triv}} \times
  \mathbb{R}(\Gamma, \pi)(M) \longrightarrow \Lambda[-(2n-1)]\ }
\]

\[
  \mathbb{R}(\Gamma, \pi)(M) \longrightarrow
  \mathbb{R}\Gamma(W_0)^\vee[-(2n-1)]^{\mathrm{triv}} \simeq \mathbb{R}\Gamma(W_0)^{\mathrm{triv}}
\]

\[
  \cdots \longrightarrow H^{i-2}(W_0) \longrightarrow H^{i}(W) \longrightarrow
  H^{i}(W - W_0) \longrightarrow H^{i-1}(W_0) \longrightarrow H^{i+1}(W)
\]

\[
  \begin{aligned}
  &\underbrace{H^{2n}(W)}_{=0} \to H^{2n}(W - W_0) \xrightarrow{\ \mathrm{inj}\ }
  \underbrace{H^{2n-1}(W_0)}_{\simeq \Lambda} \xrightarrow{\ \mathrm{surj}\ }
  \underbrace{H^{2n+1}(W)}_{=\Lambda} \\
  &\qquad \to \underbrace{H^{2n+1}(W - W_0)}_{=0}
  \to \underbrace{H^{2n}(W_0)}_{=0} \to 0
  \end{aligned}
\]

\[
  \left\{\begin{aligned}
    H^{2n}(W - W_0) &= H^{2n+1}(W - W_0) = 0 \\
    \Lambda = H^{2n-1}(W_0) &\simeq H^{2n+1}(W) = \Lambda
  \end{aligned}\right.
\]

\[
  \left\{\begin{aligned}
    H^{i}(W) \longrightarrow H^{i}(W_0) \quad &\text{un isom pour } i \leq n-2 \\
    &\text{un mono pour } i = n-1
  \end{aligned}\right.
\]

\[
  \left\{\begin{aligned}
    H^{i}_{!}(W_0) \longrightarrow H^{i+2}(W) \quad &\text{un isom si } i \geq n+1 \\
    &\text{un épim si } i = n
  \end{aligned}\right.
\]

\[
  \Bigl[\ H^{i}(W - W_0) = 0 \ \text{ si } i > n+1 \Bigr],
  \qquad H^{i}(W - W_0) = H^{i}(P).
\]

\[
  \left\{\begin{aligned}
    H^{i}(W - W_0) &= 0 \quad \text{si } i > n+1 \\
    H^{i}(W_0) &= 0 \quad \text{si } i \neq n-1, n, 0, 2n-1
  \end{aligned}\right.
\]

\[
  0 \to H^{0}(W) \to H^{0}(W - W_0) \to 0, \qquad
  0 \to H^{1}(W - W_0) \to H^{0}(W_0) \to 0
\]

\[
  \left\{\begin{aligned}
    &0 \to H^{n}(W - W_0) \to H^{n-1}(W_0) \to 0 \\
    &0 \to H^{n+1}(W - W_0) \to H^{n}(W_0) \to 0 \qquad (\text{si } n \neq 1) \\
    &0 \to H^{2n-1}(W_0) \to H^{2n+1}(W) \to 0
  \end{aligned}\right.
\]

\[
  \begin{aligned}
    &0 \to H^{0}(W) \xrightarrow{\ \sim\ } H^{0}(W - W_0) \to 0 \\
    &0 \to H^{1}(W - W_0) \to H^{0}(W_0) \to 0 \\
    &0 \to H^{2}(W - W_0) \to H^{1}(W_0) \to H^{3}(W) \simeq \Lambda \to 0 \ ]
  \end{aligned}
\]

\[
  \boxed{\ H^{i}(M, \Lambda) = 0 \quad \text{si } i > n\ }
\]

\[
  \cdots \to H^{2n-i}(M)^\vee \to H^{i}(M) \to H^{i}(W_0) \to
  H^{2n-i-1}(M)^\vee \to H^{i+1}(M) \to H^{i+1}(W_0)
\]

\[
  \left\{\begin{aligned}
    H^{i}(W_0) &\xrightarrow{\ \sim\ } H^{2n-i-1}(M)^\vee && \text{si } i \geq n+2
      && (\text{épim si } i = n+1) \quad (\text{donc } 2n-i-1 \leq n-3) \\
    H^{i}(M) &\xrightarrow{\ \sim\ } H^{i}(W_0) && \text{si } i \leq n-2
      && (\text{mono si } i = n-1)
  \end{aligned}\right.
\]

\[
  \boxed{\begin{array}{c}
    0 \to H^{n+1}(M)^\vee \xrightarrow{\ u\ } H^{n-1}(M) \to H^{n-1}(W_0) \to H^{n}(M)^\vee \\
    \downarrow \\
    0 \leftarrow H^{n+1}(M) \xleftarrow{\ {}^t u\ } H^{n-1}(M)^\vee \leftarrow H^{n}(W_0) \leftarrow H^{n}(M)
  \end{array}}
\]

\[
  \Updownarrow
\]

\[
  \left\{\begin{aligned}
    &H^{i}(V_\eta) \to H^{i-1}(V_s) \ \text{un \emph{isom} si } i \neq 0, 2n \\
    &\emph{et}\ \left\{\begin{aligned}
      &H^{0}(V_\eta) \simeq \Lambda \\
      &H^{2n}(V_\eta) \to H^{2n-1}(V_s) \ \text{inj et conoyau} \simeq \Lambda \\
    \end{aligned}\right. \qquad (\text{quand } n \geq 1)
  \end{aligned}\right.
\]

\[
  \Bigl[\ 0 \to \underbrace{H^{2n}(V_\eta)}_{=0 \text{ si } n \geq 2} \to H^{2n-1}(V_s)
  \to H^{2n+1}(V) \to \underbrace{H^{2n+1}(V_\eta)}_{=0 \text{ si } n \geq 1}
  \to \underbrace{H^{2n}(V_s)}_{=0} \to 0 \ \Bigr]
\]

\[
  \Downarrow \quad \text{si on suppose déjà que l'on a singularité « isolée »}
\]

\[
  H^{i}(V_s) = 0 \quad \text{si } i \neq 0, n-1, n
\]