  \documentclass[11pt]{amsart}
%SetFonts
% mathpazo
\usepackage{amsmath}
%\usepackage[noBBpl,sc,osf]{mathpazo} % math & rm
\linespread{1.05} % Palatino needs more leading (space between lines)
\usepackage[scaled]{helvet} % ss
%\usepackage{zlmtt} % tt
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{amssymb}
\usepackage{color}
%SetFonts
\newcommand{\colr}{\color{blue}}
%     If your article includes graphics, uncomment this command.
\usepackage{graphicx}
\usepackage{tikz}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{csquotes}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usepackage{csquotes}
\usepackage{enumerate}
%\usepackage{chngcntr}
%\usepackage{apptools}
%\AtAppendix{\counterwithin{lemma}{section}}
%\AtAppendix{\counterwithin{theorem}{section}}

\usepackage{amsmath, amsfonts,amsthm,graphicx,fullpage,etex,tikz,hyperref}



\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=cyan,
}
%\usepackage{amsfonts}
\usepackage{amssymb}



%\usepackage{todonotes, pictex}
%\newcommand{\tinytodo}[2][]
%{\todo[caption={#2}, #1]{\renewcommand{\baselinestretch}{0.5}\selectfont#2\par}}

%\usepackage{ulem}

\paperwidth=\dimexpr \paperwidth + 3cm\relax
\oddsidemargin=\dimexpr\oddsidemargin + 2cm\relax
\evensidemargin=\dimexpr\evensidemargin + 2cm\relax
\marginparwidth=\dimexpr \marginparwidth + 1cm\relax

%\usepackage{biblatex}
%style=alphabetic,
%orting=nty
%]{biblatex}
\usepackage{enumerate}
%\addbibresource{all.bib} 
%\bibliography{all.bib}

\renewcommand{\thepart}{\Roman{part}}


\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{Definition}[theorem]{Definition}


\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{partremark}{Remark}[part]
\newtheorem{notation}[theorem]{Notations}
\newtheorem{StandingAssumption}[theorem]{Standing Assumption}
\newtheorem*{Obstacle}{Obstacle}
\newtheorem*{claim}{Claim}
\newtheorem*{comment}{comment}
\newtheorem*{strategy}{Basic Strategy}

\theoremstyle{definition}
\newtheorem{problem}{Problem}
\newtheorem{definition}[theorem]{Definition}

\numberwithin{equation}{section}

\newcommand{\T}{\mathbb{T}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\Aff}{\mathrm{Aff}}
\newcommand{\Aut}{\mathrm{Aut}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\id}{\mathrm{Id}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\s}{\mathcal{S}}
\newcommand{\n}{\mathbf{n}}
\newcommand{\dist}{\mathrm{dist}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Vol}{\mathrm{Vol}}
\newcommand{\GL}{\mathrm{GL}}
\def\cg{\color{darkcyan}}
\newcommand{\mc}{\mathcal}
\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\mf}{\mathfrak}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\norm}[1]{\left|\left| #1 \right|\right|}
\newcommand{\ve}{\varepsilon}
\newcommand{\of}{\circ}
\newcommand{\tth}{^{\mbox{\normalfont\tiny th}}}
\newcommand{\mbf}{\mathbf}

\newcommand{\holder}{H\"{o}lder }
\newcommand{\smooth}{ C^{\infty}}
\newcommand{\noi}{\noindent}
\newcommand{\td}{\todo[inline]}
\definecolor{darkcyan}{rgb}{0. 0.65, 0.65}
\def\cb{\color{blue}}
\def\h-rk{rk ^{h}}
\def\hrk{rk ^{h}}
%\hoffset=-1.5cm


\newtheorem*{res}{Theorem}
%\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{prob}[theorem]{Problem}
\newtheorem{Structural Stability Theorem}[theorem]{Structural Stability Theorem}

\def\bt{\begin{theorem}}
\def\et{\end{theorem}}
\def\bd{\begin{definition}}
\def\ed{\end{definition}}
\def\bl{\begin{lemma}}
\def\el{\end{lemma}}
\def\holder{ H\"{o}lder }
\def\Sym{\operatorname{Sym}}
\def\Diff{\operatorname{Diff}}
\def\Homeo{\operatorname{Homeo}}
\def\Lie{\operatorname{Lie}}
\def\Stab{\operatorname{Stab}}
\def\loc{\operatorname{loc}}
\def\Heis{\operatorname{Heis}}
\def\rank{\operatorname{rank}}
\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\diag{\operatorname{diag}}
\def\vol{\operatorname{vol}}
\def\Isom{\operatorname{Isom}}
\def\Fix{\operatorname{Fix}}
\def\Int{\operatorname{Int}}
\def\Per{\operatorname{Per}}
\def\Gr{\operatorname{Gr}}
\def\Ann{\operatorname{Ann}}
\def\del{\partial}

\title{Smooth Manifolds Fall 2023 - Homework 2}

\begin{document}

\maketitle



\begin{problem}
Consider the transformation group $\Gamma = \set{ R_{k\pi/2} : k \in \Z}$, where $R_\theta : \R^2 \to \R^2$ is the linear transformation determined by the matrix $R_\theta = \begin{pmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix} $. Show that the quotient space $\R^2 / \Gamma$ is a topological manifold. Is it homeomorphic to a familiar one? Furthermore, does it have a smooth structure such that the map $\pi : \R^2 \to \R^2 /\Gamma$ is a submersion? Does there exist a smooth structure such that $\pi$ is $C^\infty$? Justify your answers through pictures and ``moral'' arguments: do not write formal proofs.
\end{problem}

\begin{problem}
Let $\mathbb{T}^2 = \R^2 / \Z^2$ be the 2-torus, and assume that $D \subset \mathbb{T}^2$ is a discrete subgroup. Show that $\mathbb{T}^2 / D$ is diffeomorphic to $\T^2$. [{\it Hint}: Show that the lift of $D$ to $\R^2$ is also a discrete subgroup containing $\Z^2$. You may use the fact that all discrete subgroups of $\R^2$ are isomorphic to $\{0\}$ $\Z$, and $\Z^2$.]
\end{problem}

\begin{problem}
Let $\varphi : U \to \R^n$ be a smooth chart for a smooth manifold $M$. Define a corresponding set $\hat{U} = \bigcup_{p \in U} T_pM \subset TM$, and let an element of $\hat{U}$ be denoted by $v_p$, where $p$ denotes the basepoint of the vector $v$. Define $\hat{\varphi} : \hat{U} \to \R^n \times \R^n$ by

\[ \hat{\varphi}(v_p) = (\varphi(p),v_{p,\varphi}),\]

where $v_{p,\varphi} \in T_p^\varphi M := T_{\varphi(p)}\R^n = \R^n$ is the vector $v$ as represented in the chart $\varphi$ (recall that $T_pM$ is isomorphic to each $T_p^\varphi M$ for any chart $\varphi$ whose domain contains $p$). Show that if $\mc A$ is a smooth atlas of charts for $M$, then $\hat{\mc A} = \set{(\hat{U},\hat{\varphi}) : (U,\varphi) \in \mc A}$ is a smooth atlas on $TM$ (this is the {\it smooth atlas on $TM$ induced by $\mc A$}).
\end{problem}

\begin{problem}
Show that if $U \subset \R^k$ is open, $\varphi : U \to \R^n$ is $C^\infty$ and $d\varphi(x)$ is injective, then there exists a $C^\infty$ change of coordinates diffeomorphism $H : \R^n \to \R^n$ defined near $x$ such that $H \of \varphi$ takes values in $\R^k \subset \R^n$.
\end{problem}

\begin{problem}
Let $M$ be a $C^\infty$ manifold, $k \in \N$ and $X \subset M$ be a closed, connected subset such that for every $x \in X$, there exists $U \subset \R^k$ and an embedding $\varphi : U \to M$ such that the image of $\varphi$ is an open neighborhood of $x$ in $X$. Show that $X$ has a $C^\infty$ manifold structure such that the inclusion of $X$ into $M$ is an embedding. [{\it Hint}: Use the previous problem!] 
\end{problem}


%\begin{problem}
%Show that if $M$ is a smooth manifold, then $\tilde{M}$, its universal cover, has a unique smooth structure such that if $\Gamma$ denotes the group of deck transformations, then the group action $\Gamma \curvearrowright \tilde{M}$ is by $C^\infty$ diffeomorphisms and the canonical homeomorphism between $\tilde{M} / \Gamma$ and $M$ is a diffeomorphism.
%\end{problem}

%\begin{problem}
%Recall that if $M$ is a smooth manifold, an $\ell$-dimensional foliation $\mc F$ of $M$ is a smooth atlas of charts (contained in the smooth structure on $M$) whose transition maps are {\it equivariant} with respect to the affine spaces $\R^{\ell} \times \set{x}$, $x \in \R^{n-\ell}$: $\psi \of \varphi^{-1}(\R^{\ell} \times \set{x}) \subset \R^{\ell} \times \set{\psi(\varphi^{-1}(x))}$. 

%We say that $y$ is {\it in the same leaf as $x$} if there exists finitely many charts $\varphi_1,\dots,\varphi_k$ and points $x = x_0, x_1,x_2,\dots,x_k = y$ such that $x_{i-1}$ and $x_i$ both belong to the domain of $\varphi_i$, and $\pi(\varphi_i(x_{i-1})) = \pi(\varphi_i(x_i))$, where $\pi$ is the projection of $\R^n$ onto its last $n-\ell$ coordinates. Show that if $L(x)$ denotes the set of all points in the same leaf as $x$, then $L(x)$ is an immersed submanifold. [{\it Note}: You have to first build the smooth manifold structure on $L(x)$ abstractly, then prove that its inclusion into $M$ is an immersion. The hard part is building the topology, the subspace topology won't (always) work!]
%\end{problem}

\end{document}




