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\usepackage{enumerate}
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\newtheorem{exercise}[]{Exercise}
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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\lra}{\longrightarrow}

\begin{document}
\pagestyle{fancy}
\lhead{1300F Geometry and Topology, Assignment 4}
\rhead{Due date: November 24, 2017}
\cfoot{}


\begin{exercise}
Let $X$ be compact and $f:X\lra Y$ smooth with $\dim X = \dim Y$ and
$Y$ connected. Recall that the mod 2 degree of $f$ is defined in terms of the mod 2 intersection number as follows:  $\deg_2(f) =
I_2(f,\iota)$, where $\iota:y\mapsto Y$ is the inclusion map of a point $y\in Y$.  
\begin{enumerate}
\vspace{-5pt}\item Prove that $\deg_2(f)$ is independent of the point $y\in Y$.  

\vspace{-5pt}\item A map $f:X\lra Y$ is called \emph{essential} when it is not homotopic to a constant map.  Prove that if $\deg_2(f)=1$, then $f$ is essential. 

\vspace{-5pt}\item Can there exist a smooth map $f:S^2\lra T^2$ with $\deg_2(f)=1$?  [Hint: consider two embedded circles $C_1,C_2$ in $T^2$ intersecting transversally at a single point.]  Can there exist a smooth map of $\deg_2(f)=1$ in the opposite direction? In each case, give proofs.
\end{enumerate}
\end{exercise}




\begin{exercise}
Let $f:S^1\lra \RR^2$ be an embedding and choose $p\in \RR^2\backslash f(S^1)$.  Define $f_p: S^1\lra S^1$ by $f_p(z) = \frac{f(z)-p}{|f(z)-p|}.$
Then we define the mod 2 winding number of $f$ about $p$ to be the degree of $f_p$, i.e. $w_2(f,p) = \deg_2(f_p)$.  
\begin{enumerate}
\item Compute $w_2(f,p)$ for the standard embedding of $S^1$ in $\RR^2$, and for any $p$. 
\item Let $R_p(v)$ be the ray starting at $p$ with direction $v\in S^1$.  Prove that $v\in S^1$ is a critical value of $f_p$ if and only if $R_p(v)$ is somewhere tangent to $f(S^1)$.
\item Show that $w_2(f,p)$ coincides with the number of points mod 2 in $R_p(v)\cap f(S^1)$, whenever $v$ is a regular value of $f_p$.
\item Show that there are points $p,q\in\RR^2\backslash f(S^1)$ such that $w_2(f,p) = 0$ and $w_2(f,q)=1$.  Show that this implies that $\RR^2\backslash f(S^1)$ has at least two components. 
%\item Fix $a\in f(S^1)$.  Show that it is possible to choose a coordinate chart $(U,\varphi)$ containing $a$ such that $\varphi(U)$ contains $(-2,2)\times (-2,2)$, $\varphi(a) = (0,0)$, and $\varphi(U\cap f(S^1)) = \{(x,y)\ :\ y=0\}$. \item Prove that each point $p\in \RR^2\backslash f(S^1)$ may be connected by a continuous path to either $\varphi^{-1}(0,1)$ or $\varphi^{-1}(0,-1)$. [Hint:  recall the tubular neighbourhood theorem].  Conclude that $\RR^2\backslash f(S^1)$ has two connected components. This is the smooth Jordan theorem. 
\end{enumerate}

\end{exercise}



\begin{exercise}
Consider an immersion $i:S^1\to \RR^2$ following a ``figure eight path'' as shown below.
\begin{center}
\begin{tikzpicture}
\draw[color=black,domain=0:.5,samples=500,smooth] plot (canvas polar cs:angle=\x r,radius= {100*sin(2*3.1415*\x r)});    %r = angle en radian
\draw[color=black,domain=0:.5,samples=500,smooth] plot (canvas polar cs:angle=\x r,radius= {-100*sin(2*3.1415*\x r)});
    %\draw[->] (-2,0) -- (2,0);
    %\draw[->] (0,-2) -- (0,2);
    %\draw[->,thick] (0,0) arc [radius=1, start angle=180, end angle= 120];
\end{tikzpicture}
\end{center}
Prove that there is no smooth homotopy from $i$ to the standard embedding $j:S^1\to \RR^2$ which remains an immersion at all intermediate times. \end{exercise}


\begin{exercise}
  Let $f:M\to\RR$ be a proper submersion.  Then $V = \ker Df$ defines
  a codimension 1 subbundle of $TM$ called the vertical bundle.
\begin{enumerate}
\item Show, using a partition of unity, that it is possible to choose
  a rank 1 subbundle $H\subset TM$ complementary to $V$.  Do not use a
  Riemannian metric.
\item Conclude that to any vector field $v$ on $\RR$ we may associate
  a unique vector field $v^{h}$ on $M$ which lies in $H$.  This is
  called the horizontal lift of $v$.
\item Prove the preimages of any pair of points in the image of $f$
  are diffeomorphic manifolds.
\end{enumerate}
\end{exercise}



\end{document}
