  \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{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}
\usepackage{enumitem}
%\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\Hom{\operatorname{Hom}}

\title{Smooth Manifolds Fall 2023 - Homework 1}

\begin{document}

\maketitle

In the following problems, a ``fake $n$-manifold'' is a topological space $M$ which is locally Euclidean. That is, for every point $p \in M$, there exists a neighborhood $U \subset M$ of $p$, an open set $V \subset \R^n$ and a homeomorphism $\varphi : U \to V$.

\begin{problem}
Let $M$ be a connected topological $n$-manifold, and $C \subset M$ be a closed proper subset. Let $M^f$ be the set

\[ M^f = \left\lbrace{ (x,a) : x \in M, \mbox{ and } \begin{array}{ll} a = 0, & x \not\in C \\ a = \pm 1, & x \in C \end{array}}\right\rbrace\]

Define a topology on $M^f$ as being generated by two types of open sets from derived from the topology on $M$: If $U \subset M$ is open, let 
\begin{itemize}
\item $U^\# = \set{(x,0) : x \in U\setminus C} \cup \set{(x,1) : x \in C\cap U}$, and \\
\item $U^\flat = \set{(x,0) : x \in U \setminus C} \cup \set{(x,-1) : x \in C \cap U}$.
\end{itemize}
 Show that, with the topology generated by sets of the form $U^\sharp$ and $U^\flat$, $M^f$ is a fake $n$-manifold, but not a manifold. What happens if $C = M$?
\end{problem}

\begin{remark}In the previous problem, when $M = \R$ and $C = \set{0}$, this is often called the ``line with two origins.''\end{remark}

\begin{problem}
Give an example of a Hausdorff fake $n$-manifold which is not a manifold (and justify why it is not a manfiold). [{\it Hint}: A disjoint union of locally Euclidean spaces is still locally Euclidean] 
\end{problem}

\begin{remark}
The example you come up with in the previous problem is probably not connected. For a connected example, look up a pathology called the \href{https://en.wikipedia.org/wiki/Long_line_(topology)}{\it long line}.
\end{remark}

%\begin{problem}
%Show that any open subset of $\R^n$ is a manifold. Find a canonical smooth structure on it.
%\end{problem}

\begin{problem}
Show that $S^n = \set{x \in \R^{n+1} : \norm{x} =1}$  has a canonical smooth $n$-manifold structure by explicitly finding a smooth atlas and showing the atlas is smooth.
\end{problem}

%\begin{problem}
%Let $M$ be a smooth $n$-manifold, $U$ and $V$ be subsets of $M$ which are domains of chart maps $\varphi$ and $\psi$, respecctively. Show that if $f : M \to \R$ is a continuous function, then $f \of \varphi^{-1}$ is $C^\infty$ on $\varphi(U \cap V)$ if and only if $f \of \psi^{-1}$ is $C^\infty$ on $\psi(U \cap V)$.
%\end{problem}

\begin{problem}
Show that if $M$ is a smooth $m$-manifold and $N$ is a smooth $n$-manifold, then $M \times N$ has a canonical smooth $(m+n)$-manifold structure.
\end{problem}

\begin{problem}
Show that $\R^2 \setminus \set{0}$, $A = \set{ x \in \R^2 : 1 < \norm{x} < 2}$ and $S^1 \times (0,1)$ are all diffeomorphic with their standard smooth structures. [{\it Hint}: Find explicit diffeomorphisms between them. Show that the maps you find are bijective and differentiable, with invertible derivative.]
\end{problem}

\begin{problem}
Let $X$ denote the boundary of the unit square in $\R^2$. Prove or find a counterexample:

\begin{enumerate}
\item $X$ is a topological 1-manifold.
\item There exists a smooth structure on $X$.
\item There exists a smooth structure on $X$ such that the inclusion of $X$ into $\R^2$ is $C^\infty$.
\item There exists a smooth structure on $X$ such that the inclusion of $X$ into $\R^2$ is an immersion.
\end{enumerate}
\end{problem}

%\begin{problem}
%Show that if $M_1$, $M_2$ and $M_3$ are smooth, connected $n$-manifolds and there exist $C^\infty$ maps $\varphi : M_1 \to M_2$, $\psi : M_2 \to M_3$ and $\tau : M_3 \to M_1$ such that $\tau \of \psi \of \varphi = \id$, then $M_1$, $M_2$ and $M_3$ are all diffeomorphic.
%\end{problem}

\pagebreak

\noindent {\bf \Large Linear algebra and vector calculus review}

\begin{problem}
Let $F : \R^n \to \R^n$ be a diffeomorphism, and assume that there exists a $v \in \R^n$ such that $v$ is an eigenvector of $dF(x)$ with real eigenvalue for every $x \in \R^n$. Show that the lines $L(x) = \set{x + tv : t \in \R}$ are {\it equivariant}: $F(L(x)) = L(F(x))$.
\end{problem}

\begin{problem}[Contraction mapping principle, differentiable version]
Let $F : \R^n \to \R^n$ be a diffeomorphism such that the eigenvalues of $DF(x)$ all have modulus at most some $\lambda < 1$ for every $x \in \R^n$. Show that $F$ has a unique fixed point $x_0$, and for every $x \in \R^n$, $F^k(x) \to x_0$ as $k \to \infty$.
\end{problem}

\begin{problem}
Let $V$ and $W$ be (real) finite-dimensional vector spaces and $\Hom(V,W)$ be the set of linear transformations from $V$ to $W$. 

\begin{enumerate}
\item Show that $\Hom(V,W)$ is a real vector space.
\item With fixed bases for $V$ and $W$, find an isomorphism between $\Hom(V,W)$ and $M(m,n)$, the set of $m \times n$ matrices, where $m = \dim(V)$ and $n = \dim (W)$.
\item If $V_0 \subset V$ is a subspace of $V$, let $\Ann(V_0) \subset \Hom(V,W)$ be the {\it annihilator} of $V_0$. That is, the set of $\varphi \in \Hom(V,W)$ such that $\varphi(v) = 0$ for all $v \in V_0$. Show that $\Ann(V_0)$ is a vector subspace of $\Hom(V,W)$, then find and prove a formula for $\dim(\Ann(V_0))$ in terms of $\dim(V)$, $\dim(W)$ and $\dim(V_0)$. [{\it Hint}: It might be useful to think about it as matrices using the previous part]
\item* Find a canonical isomorphism between $V^* \otimes W$ and $\Hom(V,W)$, and prove it is an isomorphism. Construct a projection $\pi : V^* \otimes W \to V_0^* \otimes W$ such that $\Ann(V_0) = \ker  \pi$, and prove that it is a projection, and that the kernel is as described. Deduce the formula for $\dim(\Ann(V_0))$ using $\pi$, as well.
\end{enumerate}
\end{problem}



\end{document}




