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\begin{document}

\thispagestyle{fancy}
\lhead{Advanced Geometry 2017--2018}
\rhead{ENS de Lyon}
\begin{center}
\Large{Reminder on topology and calculus}
\end{center}
\vspace{-7mm}
\rule{\linewidth}{0.5pt}

\begin{exo}[Calculus]
Show that the following maps are differentiable, and compute their differentials.
\begin{enumerate}
\item $(A,B) \longmapsto AB$ from $\mathcal{M}_{nk}(\R) \times \mathcal{M}_{kp}(\R)$ to $\mathcal{M}_{np}(\R)$.
\item $\det : \mathcal{M}_n(\R) \to \R$.
\item \label{3} $f \mapsto f^{-1}$ from $GL(E)$ to itself, where $E$ is a real vector space of finite dimension.
\item Let $\Omega \subset \R^n$ be an open set of compact adherence and $V$ a vector space of finite dimension of functions $\R^n \to \R$ of class $\mathcal{C}^1$.
 We consider $F:(f,x) \mapsto f(x)$ from $V\times \Omega$ to $\R$.
\end{enumerate}
\end{exo}

\begin{exo}[Submanifolds]
\label{sousvar}
\begin{enumerate}
\item Recall the four equivalent definitions of a $d$-dimensional submanifold of $\R^n$ (without proving that they are equivalent).
\item Among the following sets, which ones are submanifolds of $\R^n$? Give their dimensions. No precise justification is expected.
\begin{enumerate}
\item The sphere of radius 1 in $\R^n$ for the euclidean norm.
\item The sphere of radius 1 in $\R^n$ for the sup norm.
\item A disjoint union of a straight line and a plane in $\R^3$.
\item  $\{(x,y) \in \R^2 \mid x^2-y^2 =0\}$.
\item  $\{(x,y) \in \R^2 \setminus \{0\} \mid x^2-y^2 =0\}$.
\item The image of $h:]-\infty,1[ \ \to \R^2$ with $h:t \mapsto \left(\frac{t^2-1}{t^2+1},t\frac{t^2-1}{t^2+1}\right)$.
\end{enumerate}
\item Let $\Omega$ be an open subset of $\R^d$ and $h:\Omega \to \R^n$ an injective immersion, is $h(\Omega)$ a submanifold of $\R^n$?
\end{enumerate}
\end{exo}

\begin{exo}[Classical groups of matrices]
Show that the following subgroups $\mathcal{M}_n(\R)$ are submanifolds of $\mathcal{M}_n(\R)$, and give their respective dimensions.
\begin{enumerate}
\item $GL_n(\R)$,
\item $SL_n(\R)$ (subgroup of matrices of determinant $1$),
\item $O_n(\R)$ (subgroup of orthogonal matrices).
\end{enumerate}
\end{exo}

\begin{exo}[A submanifold is a manifold]
Let $\Sigma$ be a submanifold of dimension $d$ of $\R^n$ with $d \leq n$ and let $\iota : \Sigma \rightarrow \R^n$ be the canonical injection from $\Sigma$ into $\R^n$.
\begin{enumerate}
\item Show that $\Sigma$ is a manifold.
\item Let $f$ be a smooth map from $\R^n$ to $\R$. Show that the restriction $f_{/\Sigma}:\Sigma \to \R$ is smooth.
\end{enumerate}
\end{exo}

\begin{exo}[Quotient topology]
\label{quotient}
\begin{enumerate}
\item Let $X$ be a topological space and $\sim$ be an equivalence relation on $X$. We denote by $p : X \to X/\sim$ the canonical projection. Recall the definition of the quotient topology on $X/\sim$?
\item Let $f:X/\sim \ \to Y$. Show that $f$ is continuous if and only if $f \circ p$ is.
\item Let $\T^n = \R^n / \Z^n$ be the $n$-dimensional torus. Show that $\T^n$ is compact and Hausdorff, and that $p$ is an open map.
\item Let $f:K \to Y$ be continuous and bijective with $K$ compact Hausdorff and $Y$ Hausdorff. Show that $f$ is an homeomorphism? Give a counter-example if $Y$ is not Hausdorff.
\item We define $\S^1$ as $\{z \in \C \mid \norm{z}=1\}$. Show that $\T^1$ is homeomorphic to $\S^1$. More generally, show that $\T^n$ is homeomorphic to $(\S^1)^n \subset \C^n$.
\item Let $\R\P^n$ be the space defined as the quotient of $\R^{n+1}\setminus \{0\}$ by the equivalence relation ``being colinear''. Show that $\R\P^n$ is compact Hausdorff and that $p$ is open.
\end{enumerate}
\end{exo}

\begin{exo}[Surface of genus $g$]
For any $g \in \N$, find a explicit map $F_g: \R^3 \to \R$ such that $\Sigma_g = (F_g)^{-1}(0)$ is a smooth surface of genus $g$ (that is a torus with $g$ holes).

\begin{figure}[h!]
\subfloat[$g=0$]{\includegraphics[width=4cm]{sphere.pdf}}
\hfill
\subfloat[$g=1$]{\includegraphics[width=4cm]{torus1.png}}
\hfill
\subfloat[$g=2$]{\includegraphics[width=4cm]{torus2.png}}
\end{figure}

\emph{Hint:} Seek $F_g$ of the form $F_g:(x,y,z)\mapsto z^2-f_g(x,y)$, where $f_g:\R^2 \to \R$ is smooth.
\end{exo}

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