\documentclass[letterpaper, 12pt]{article}
\usepackage[left=2cm,top=3cm,right=2cm]{geometry}
\usepackage{amsmath,amsfonts,amsthm,tikz}
\usepackage[all]{xy}
\usepackage{fancyhdr}
\usepackage{enumerate}
\theoremstyle{definition}
\newtheorem{exercise}[]{Exercise}
\newtheorem{bonus}{Bonus}[exercise]

% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}

\begin{document}
\pagestyle{fancy}
\lhead{1300F Geometry and Topology, Assignment 4}
\rhead{Due date: November 24, 2016}
\cfoot{}


\begin{exercise}
Fix a positive integer $k$ and complex numbers $c_0,c_1,\ldots, c_{k-1}$.  Consider the map $f:\CC\to\CC$ given by 
\[
f(z) = z^k + c_{k-1} z^{k-1} + \cdots + c_1 z + c_0.
\]
\begin{enumerate}
\item Identifying $\CC$ with the standard affine chart $U_0$ for $\CC P^1$, prove that $f$ extends uniquely to a smooth map from $\CC P^1$ to $\CC P^1$.  Call this map $\tilde f$.
\item Determine the degree modulo $2$ of $\tilde f$. 
\item If $k$ is odd, can you show that $\tilde f$ is surjective? What does this imply about the number of zeros of $f$ in this case? How is this related to the fundamental theorem of algebra? 
\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}




\begin{exercise}
Let $V$ be a finite dimensional vector space, and view it as a manifold $M$. Then $TM = V\times V$.
\begin{enumerate}
\item The trivial map $E:x\mapsto (x,x)$ defines a section of the
  tangent bundle, i.e. a vector field.  Compute the time-$t$ flow of
  this vector field and determine whether it is complete or not.
\item Suppose $A:V\to V$ is a linear map. Then the map $A:x\mapsto
  (x,Ax)$ defines a vector field on $M$; compute its flow and
  determine if it is complete.
\item If $A,B$ are linear maps as above, compute the Lie derivative of
  the vector fields they determine.  Verify the fact that if the
  vector fields commute, then the flows commute.
\end{enumerate}

\end{exercise}

\end{document}
