memoire-m2

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -87,7 +87,7 @@ strands.
   The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group
   \(\pi_1(C(\mathbb{D}^2, n), *)\) of the unordered configuration space
   \(C(\mathbb{D}^2, n) = \mfrac{C^{\operatorname{ord}}(\mathbb{D}^2,
-  n)}{\mathfrak{S}_n}\) of \(n\) distinct points in the interior of the disk.
+  n)}{S_n}\) of \(n\) distinct points in the interior of the disk.
   The elements of \(B_n\) are referred to as \emph{braids}.
 \end{definition}
 

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -207,14 +207,14 @@ Given an orientable surface \(\Sigma\) and \(x_1, \ldots, x_n \in \Sigma\degree\
 denote by \(\Mod(\Sigma \setminus \{x_1, \ldots, x_n\})_{\{x_1, \ldots,
 x_n\}} \subset \Mod(\Sigma \setminus \{x_1, \ldots, x_n\})\) the subgroup of mapping
 classes \(f\) that permute \(x_1, \ldots, x_n\) -- i.e. \(f \cdot x_i =
-x_{\sigma(i)}\) for some \(\sigma \in \mathfrak{S}_n\). We certainly have a
+x_{\sigma(i)}\) for some \(\sigma \in S_n\). We certainly have a
 surjective homomorphism \(\operatorname{forget} : \Mod(\Sigma \setminus \{x_1,
 \ldots, x_n\})_{\{x_1, \ldots, x_n\}} \to \Mod(\Sigma)\) which ``\emph{forgets} the
 additional punctures \(x_1, \ldots, x_n\) of \(\Sigma \setminus \{x_1, \ldots,
 x_n\}\),'' but what is its kernel?
 
 To answer this question, we consider the configuration space \(C(\Sigma, n) =
-\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{\mathfrak{S}_n}\) of \(n\) (unordered)
+\mfrac{C^{\operatorname{ord}}(\Sigma, n)}{S_n}\) of \(n\) (unordered)
 points in the interior of \(\Sigma\) -- where \(C^{\operatorname{ord}}(\Sigma, n) = \{
 (x_1, \ldots, x_n) \in (\Sigma\degree)^n : x_i \ne x_j \ \text{for}\ i \ne j \}\).
 Denote \(\Homeo^+(\Sigma, \partial \Sigma)_{x_1, \ldots, x_n} = \{\phi \in \Homeo^+(\Sigma,
@@ -263,7 +263,7 @@ and its long exact sequence in homotopy we then get\dots
 We may regard a simple loop \(\alpha \subset C(\Sigma, n)\) based at \([x_1, \ldots,
 x_n]\) as \(n\) disjoint curves \(\alpha_1, \ldots, \alpha_n \subset \Sigma\) with
 \(\alpha_i(0) = x_i\) and \(\alpha_i(1) = x_{\sigma(i)}\) for some \(\sigma \in
-\mathfrak{S}_n\). The element \(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) can
+S_n\). The element \(\operatorname{push}([\alpha]) \in \Mod(\Sigma)\) can
 then be seen as the mapping class that ``\emph{pushes} a neighborhood of
 \(x_{\sigma(i)}\) towards \(x_i\) along the curve \(\alpha_i^{-1}\),'' as shown
 in Figure~\ref{fig:push-map} for the case \(n = 1\). Indeed, this goes to