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\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}} \newcommand{\del}{\partial}
\newcommand{\Cc}{\mathcal{C}}\newcommand{\Dd}{\mathcal{D}}
\newcommand{\IR}{\mathbf{I}}\newcommand{\Id}{\mathrm{id}}

\begin{document}
\pagestyle{fancy} \lhead{1301 Topology II, Assignment 1} \rhead{Due
  date: January 31, 2013} \cfoot{}
\noindent {\bf Reading:} Hatcher \S 1.1.

\begin{exercise}
  \begin{enumerate}
  \item Determine the de Rham cohomology of $\RR^3\setminus Z$, where
    $Z$ is the union of $\{x=y=0, z\geq 0\}$, $\{y=z=0, x\geq 0\}$,
    and $\{z=x=0, y\geq 0\}$.
  \item Use Mayer-Vietoris to determine the de Rham cohomology of a
    compact orientable surface of genus $g$.
  \end{enumerate}
\end{exercise}

\begin{exercise}
  Hatcher \S 1.1, exercises 13, 16, 18.
\end{exercise}

\begin{exercise}
  Let $p_0,p_1\in S^2$ be distinct points. Show that any two paths
  $\gamma_0,\gamma_1:[0,1]\to S^2\ :\ \gamma_0(i)=\gamma_1(i)=p_i,\
  i=0,1$ must be homotopic with fixed endpoints.
  Use this to describe the fundamental groupoid $\Pi S^2$ completely.
  Also use this to compute $\pi_1(\RR P^2)$.  Compute $\pi_1(\RR
  P^n)$.
\end{exercise}

\begin{exercise}
  In $\RR^3$, let $C_1$ be the z-axis, and let $C_2$ be the circle
  $\{x^2 + y^2=1\ \text{and}\ z=0\}$.  Compute
  $\pi_1(\RR^3\setminus\{C_1\cup C_2\})$ and express it in terms of
  generators and relations.  Draw a picture of the generators, and
  draw a picture of the relations.
  Use this to compute the fundamental group of
  $\RR^3\setminus\{\text{Hopf link}\}$.
\end{exercise}

% \begin{exercise}
%   Find a pointed topological space $(X,x_0)$ such that $\pi_1(X,x_0)$
%   is not discrete (as a topological group).  In your example, is it
%   countable?
% \end{exercise}

\begin{exercise}
  Let $\IR$ be the category with two objects $\{0,1\}$ and only one
  non-identity arrow $\iota:0\to 1$.  If $\Cc,\Dd$ are categories,
  then a functor $F:\Cc\times \IR\to \Dd$ is called a \emph{natural
    transformation} from the functor $f_0:\Cc\to \Dd$ to the functor
  $f_1:\Cc\to \Dd$, where $f_i(X) = F(X,i),\ i=0,1$.  Prove that
  there is a category $\mathrm{Fun}(\Cc,\Dd)$ whose objects are
  functors from $\Cc$ to $\Dd$, and whose morphisms are natural
  transformations.  What are the invertible morphisms (isomorphisms)
  in this category?

  Two categories $\Cc,\Dd$ are \emph{equivalent} when there are
  functors $f:\Cc\to \Dd$ and $g:\Dd\to\Cc$ such that $f\circ g$ is
  isomorphic to $\Id_\Dd$ and $g\circ f$ is isomorphic to $\Id_\Cc$,
  where isomorphism is in the sense above.  Give an example of two
  categories which are equivalent but which have non-bijective objects
  (consider only small categories, i.e. categories whose objects and
  morphisms are each a set).
\end{exercise}

\begin{exercise}
  A \emph{coproduct} or \emph{sum} of two objects $X_1, X_2$ in a
  category $\Cc$ is an object $P$, equipped with arrows
  $\iota_i:X_i\to P,\ i=1,2$ such that for any other object $Q$
  equipped with arrows $\nu_i:X_i\to Q$, there exists a unique arrow
  $\nu:P\to Q$ with $\nu_i = \nu\circ \iota_i$, for $i=1,2$.  We draw
  the coproduct like this:
  \[
  \xymatrix{& X_2\ar[d]^{\iota_2}\\ X_1\ar[r]_{\iota_1} & P}
  \]

  Show that if there are two coproducts of the pair $X_1,X_2$, then
  the two coproducts are canonically isomorphic.

  Show that the category of sets, topological spaces, pointed
  topological spaces, groups, (and bonus: groupoids), always have
  coproducts, i.e. for any pair of objects $X_1,X_2$, there exists an
  object which is a coproduct of $X_1,X_2$.
\end{exercise}

\begin{exercise}
  Given two objects $X_1,X_2$ in a category $\Cc$, and an object $Z$
  mapping to both $X_i$ by $f_i:Z\to X_i$, the \emph{fibered
    coproduct} (also called \emph{fibered sum} or \emph{pushout}) of
  $X_1,X_2$ over $Z$ is an object $P$ and two morphisms
  $\iota_i:X_i\to P$ such that the diagram commutes:
  \[
  \xymatrix{Z\ar[r]^{f_2}\ar[d]_{f_1} & X_2\ar[d]^{\iota_2}\\
    X_1\ar[r]_{\iota_1} & P}
  \]
  and such that $(P,\iota_1,\iota_2)$ is universal for this diagram in
  the sense that for any other set $(Q,j_1,j_2)$ fitting in the
  diagram, there must exist $u:P\to Q$ making the following diagram
  commute:
  \[\xymatrix{Z\ar[r]^{f_2}\ar[d]_{f_1} &
    X_2\ar[d]^{\iota_2}\ar[ddr]^{j_2}&\\
    X_1\ar[r]_{\iota_1}\ar[rrd]_{j_1} & P\ar[dr]^u& \\ & & Q}\]
  Show that the categories from the previous exercise always have
  fibered sums.
\end{exercise}


% \begin{exercise}
%   if $G$ is a non-abelian group, then its abelianization is
%   $G^{ab}:=G/[G,G]$, where $[G,G]$ is the subgroup generated by all
%   commutators $[g,h] = ghg^{-1}h^{-1},\ g,h\in G$.

%   Prove that the fundamental group of any knot complement must have
%   abelianization isomorphic to $\ZZ$. Recall that a knot is an
%   embedded circle in $\RR^3$.
% \end{exercise}



\end{document}
