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\newcommand{\myTitle}{HW 6, due Friday, March 15}
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\begin{enumerate}
\item On $\proj^1$, we have the orienting atlas $\mathfrak{A}=\{(U_x,h),
  (U_y,k)\}$ where
  \begin{eqnarray*}
    h\colon U_x:=\{(x,y)\in\proj^1:x\neq0\}&\to&\R\\
    (x,y)&\mapsto& y/x\\
    &&\\
    k\colon U_y:=\{(x,y)\in\proj^1:y\neq0\}&\to&\R\\
    (x,y)&\mapsto& -x/y.\\
  \end{eqnarray*}
  Define $\omega\in\Omega^1\proj^1$ whose local description with respect to $h$
  is
  \[
  \omega|_{U_x}(a) = \frac{1}{a^2+1}\,da.
  \]
  In HW 4, we saw that this completely determines $\omega$ on $\proj^1$.
  \begin{enumerate}
    \item Let $A_1=h^{-1}([-1,1])$ and $A_2=k^{-1}([-1,1])$.  Show that
      $A_1\cup A_2=\proj^1$ with intersection of measure $0$.
      \medskip

      {\sc solution:}

    \item Compute $\int_{\proj^1}\omega$ using $A_1$ and $A_2$ (following the
      definition of the integral given in class).
      \medskip

      {\sc solution:}

  \end{enumerate}

\item Let $S^1$ be the unit circle in $\R^2$ centered at the origin.  Define
  $p(t)=(\cos(t),\sin(t))$ and $q(t)=p(t+\pi)$ for $t\in(-\pi,\pi)$.  Use
  $p^{-1}$ and $q^{-1}$ as local coordinates on $S^1$.  
  \begin{enumerate}
    \item Compute the transition function $q^{-1}\circ p$.
      \medskip

      {\sc solution:}

    \item In the local coordinate $t$ provided by $p$, consider 
      $dt_{(1,0)}\in T^*_{(1,0)}S^1$.  For each angle $\alpha$ there is a
      mapping $m_{\alpha}\colon S^1\to S^1$ sending a point with angle
      $\beta$ to the point with angle $\alpha+\beta$. Define
      \[
      \omega(\alpha) = m_{\alpha}^*(dt_{(1,0)}).
      \]
      What is the $1$-form $\omega$ in the local coordinate provided by $p$?
      \medskip

      {\sc solution:}

    \item What is $\int_{S^1}\omega$?
      \medskip

      {\sc solution:}

  \end{enumerate}

\item Read the handout on the homology of simplicial complexes.  Turn
  in problems 1, 2, 3, and 5.
  \begin{enumerate}
    \item[1.] Compute the homology for the simplicial complex on $[5]$ consisting of
  all subsets of the sets $\{1,2\}$, $\{1,3\}$, $\{2,3\}$, $\{2,4\}$, $\{3,4\}$,
  and $\{5\}$.  Draw a picture.
      \medskip

      {\sc solution:}

\item[2.] Make your own (small-ish) simplicial complex and calculate its
  homology. 
      \medskip

      {\sc solution:}

\item[3.] Prove that $\partial_i\circ\partial_{i+1}=0$ for all $i$.
      \medskip

      {\sc solution:}

\item[5.] Below is a simplicial complex drawn on a sphere.  To the right is
  the same simplicial complex, projected into the face $\{1,3,4\}$.
  In all, there are $8$ two-dimensional faces, counting $\{1,3,4\}$.
  These are the top-dimensional faces.  What are the homology groups
  for this simplicial complex?  How are your results reflected in the
  geometry of the sphere. (An important fact is that it turns out
  that you would get the same homology groups for any simplicial
  complex drawn on the sphere.)
\begin{center}
\includegraphics{sphere.pdf}
\end{center}
\begin{center}{Simplicial complex on a sphere.}\end{center}
  \end{enumerate}
      \medskip

      {\sc solution:}

\end{enumerate}
\end{document}
