\documentclass[letterpaper, 12pt]{article}
\usepackage[left=2cm,top=3cm,right=2cm]{geometry}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage[all]{xy}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{patterns}
\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}}
\newcommand{\lra}{\to}
\newcommand{\del}{\partial}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}




\begin{document}
\pagestyle{fancy}
\lhead{1300F Geometry and Topology, Assignment 3}
\rhead{Due date: November 10, 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 Describe in detail the critical points and critical values of $\tilde f$ in the case $k=2$.
\end{enumerate}
\end{exercise}
%
%
%\begin{exercise}\mbox{}
%Prove that, using the definition of submanifold given in class, any submanifold $L\subset M$ inherits a smooth structure from $M$. Also show that using this smooth structure, the inclusion map $i:L\to M$ is an embedding. 
%\end{exercise}


\begin{exercise}\mbox{}
Let $K$ be a manifold with boundary where $L,M$ are without
boundary.  Assume that $f:K\lra M$ and $g:L\lra M$ are smooth maps
such that both $f$ and $\del f$ are transverse to $g$.  Prove that the
fiber product $K\times_M L$ is a manifold with boundary equal to
$\del K\times_M L$.
\end{exercise}

% Define a variable as a length
\newcommand{\nvar}[2]{%
    \newlength{#1}
    \setlength{#1}{#2}
}

% Define a few constants for drawing
\nvar{\dg}{0.3cm}
\def\dw{0.25}\def\dh{0.5}
% Define commands for links, joints and such
\def\link{\draw [double distance=1.5mm, very thick] (0,0)--}
\def\joint{%
    \filldraw [fill=white] (0,0) circle (5pt);
    \fill[black] circle (2pt);
}
\def\grip{%
    \draw[ultra thick](0cm,\dg)--(0cm,-\dg);
    \fill (0cm, 0.5\dg)+(0cm,1.5pt) -- +(0.6\dg,0cm) -- +(0pt,-1.5pt);
    \fill (0cm, -0.5\dg)+(0cm,1.5pt) -- +(0.6\dg,0cm) -- +(0pt,-1.5pt);
}

\def\robotbase{%
    %\draw[rounded corners=8pt] (-\dw,-\dh)-- (-\dw, 0) --
    %    (0,\dh)--(\dw,0)--(\dw,-\dh);
    %\draw (-0.5,-\dh)-- (0.5,-\dh);
    %\fill[pattern=north east lines] (-0.5,-1) rectangle (0.5,-\dh);
	\fill[black] circle (6	pt);
}

% This macro draws a three link manipulator.
% Input parameters:
%   #1 theta_1
%   #2 L_1
%   #3 theta_2
%   #4 L_2
%   #5 theta_3
%   #6 L_3
%
% Example:
%   \threelink{60}{2}{-70}{2}{30}{1}
\newcommand{\twolink}[4]{%
    \link(#1:#2);
    \joint
    \robotbase
    \begin{scope}[shift=(#1:#2), rotate=#1]
        \link(#3:#4);
        \joint
        \begin{scope}[shift=(#3:#4), rotate=#3]
            \joint
        \end{scope}
    \end{scope}
}
\newcommand{\threelink}[6]{%
    \link(#1:#2);
    \joint
    \robotbase
    \begin{scope}[shift=(#1:#2), rotate=#1]
        \link(#3:#4);
        \joint
        \begin{scope}[shift=(#3:#4), rotate=#3]
            \link(#5:#6);
            \joint
            \begin{scope}[shift=(#5:#6), rotate=#5]
                \joint%\grip
            \end{scope}
        \end{scope}
    \end{scope}
}

\newcommand{\fourlink}[8]{%
    \link(#1:#2);
    \joint
    \robotbase
    \begin{scope}[shift=(#1:#2), rotate=#1]
        \link(#3:#4);
        \joint
        \begin{scope}[shift=(#3:#4), rotate=#3]
            \link(#5:#6);
            \joint
            \begin{scope}[shift=(#5:#6), rotate=#5]
                \link(#7:#8);
                \joint
                \begin{scope}[shift=(#7:#8), rotate = #7]
                \joint
                \end{scope}
               %\grip
            \end{scope}
        \end{scope}
    \end{scope}
}

\centering
\begin{tikzpicture}
    \twolink{60}{2}{-90}{2}{-60}{1}
    \begin{scope}[xshift=5cm]
        \threelink{75}{2}{-145}{2}{90}{2}
    \end{scope}
    \begin{scope}[xshift=10cm]
        \fourlink{15}{2}{-45}{2}{76}{2}{130}{2}
    \end{scope}
    
%    \begin{scope}[shift={(2cm, -3.2cm)}]
%        % Illustration of two different solutions to the inverse kinematic
%        % problem.
%        \begin{scope}[dashed]
%            \threelink{-10}{2}{70}{2}{-40}{1}
%        \end{scope}
%        \threelink{60}{2}{-70}{2}{30}{1}
%    \end{scope}
\end{tikzpicture}


\begin{exercise}\mbox{}
A planar arm consists of a number $n$ of rigid unit length segments in a plane, joined end-to-end, starting with an end with fixed position and ending with a free end, as shown above.
\begin{enumerate}
\item Give a bijection from the space of configurations $X$ of such a planar arm with n segments to $T^n = (S^1)^n$.
\item If the fixed end is located at $(0,0)\in\RR^2$, define the map $f:X\to \RR^2$ on any configuration by taking the position of its free end. Describe precisely the critical points and critical values of this map.  Give examples in the cases $n=2,3,4$.  (Using $\CC$ instead of $\RR$ might be preferred by some, this is fine) 
\item Consider the configurations $Y\subset X$ where the free end is constrained to lie on the positive x-axis $\{(x,y)\ :\ x>0, y=0\}$.  Prove that $Y$ is a smooth submanifold. 

\item On this submanifold $Y\subset X$, consider the function $g:Y\to\RR$ given by the distance of the free end from the fixed end.  Describe carefully the critical points and critical values of this map in the cases $n=2, 3, 4$.   

\item  In the case $n=3$, prove that $g^{-1}(2)$ is diffeomorphic to $S^1$.    

\item  In the case $n=4$, prove that $g^{-1}(3)$ is diffeomorphic to $S^2$. Challenge: what is $g^{-1}(1)$? 
\end{enumerate} 

\end{exercise}



\begin{bonus}\mbox{}
  Let $K,L$ be submanifolds of a manifold $M$, and suppose that their
  intersection $K\cap L$ is also a submanifold.  Then $K,L$ are said
  to have \emph{clean} intersection when, for each $p\in K\cap L$, we
  have $T_p(K\cap L) = T_pK\cap T_pL$.  Show that there are
  coordinates near $p\in K\cap L$ such that $K$, $L$, and $K\cap L$
  are given by linear subspaces of $\RR^n$ of the form
  $V(x^{i_1},\ldots, x^{i_k})$ for some subset of the coordinates.  It
  is useful to use the algebraic geometry notation $V(x^1,\ldots,
  x^k)$ to mean the ``vanishing'' subspace $x^1 = \cdots = x^k = 0$.

%  Also, can the intersection of submanifolds be transverse but not
%  clean? Can it be clean but not transverse? Give examples or proofs
%  as necessary.
\end{bonus}

\end{document}
