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

\begin{document}
\pagestyle{fancy} \lhead{1300F Geometry and Topology, Assignment 5}
\rhead{Due date: December 8, 2017} \cfoot{} 

\begin{exercise}
  Consider $S^n$ and its two stereographic coordinate charts
  $\varphi_{S},\varphi_N$ to $\RR^n$.  Using standard coordinates on
  $\RR^n$, write down the coordinate expressions for a smooth,
  nowhere-vanishing $n$-form on $S^n$ with integral equal to 1. 
\end{exercise}

\begin{exercise}
On $M = \RR^2\setminus\{0\}$ with coordinates $(x,y)$, consider the differential form 
\[
\omega = \frac{x dy - y dx}{x^2+y^2}.
\]
Prove that $\omega$ is closed, but not exact. 
\end{exercise}


\begin{exercise}
Let $\varphi:\RR^3\to \RR^3$ be given by 
\[
(r,\phi,\theta)\mapsto (r\sin\phi\cos\theta, r\sin\phi\sin\theta,r\cos\phi),
\]
where $(r,\phi,\theta)$ are standard Cartesian coordinates on $\RR^3$.
\begin{itemize}
 \item Compute $\varphi^*dx, \varphi^*dy, \varphi^*dz$ where $(x,y,z)$ are Cartesian coordinates for $\RR^3$.
\item Compute  $\varphi^*(dx\wedge dy\wedge dz)$.
\item For any vector field $X$, define $i_X$ to be the unique degree $-1$ (i.e. it reduces degree by 1) derivation (i.e. $i_X(\alpha\wedge\beta) = i_X(\alpha)\wedge \beta + (-1)^{|\alpha|}\alpha\wedge i_X(\beta)$)  of the algebra of differential forms such that $i_X(f) = 0$ and $i_Xdf = X(f)$  for $f\in\Omega^0(M)$.  Compute the integral
\[
\int_{S^2} \iota_r^*(i_X (dx\wedge dy\wedge dz)),
\] 
for the vector field $X = \varphi_* \tfrac{\del}{\del r}$, where $\iota_r:S^2\to \RR^3$ for $r>0$ fixed is defined to be the embedding of $S^2$ as the sphere of radius $r$, i.e. $\iota_r(x) = rx$.
\end{itemize}
\end{exercise}

\begin{exercise} Use Stokes' theorem if necessary:
    \noindent
    \begin{enumerate}
    \item Let $M$ be a compact orientable smooth $n$-manifold (without
      boundary) and let $\mu\in\Omega^{n-1}(M)$.  Prove there exists a
      point $p\in M$ with $d\mu(p)=0$.
    \item For any sphere $S^k$, let $\iota:S^k\to \RR^{k+1}$ be the
      usual inclusion, and let $v_k\in \Omega^k(S^k)$ be given by
        \[
        v_k = \iota^*\sum_{i=0}^k (-1)^i x^i dx^0\wedge\cdots \wedge
        dx^{i-1}\wedge dx^{i+1}\wedge\cdots\wedge dx^k.
        \]
        Show that $v_k$ is closed and that $[v_k]\neq 0$ in the top de
        Rham cohomology group $H^k(S^k)$.
       \end{enumerate}
\end{exercise}



\pagebreak
\begin{exercise}
Compute the de Rham cohomology groups (Using Mayer-Vietoris if necessary) of the following spaces, for all degrees.  
 \begin{itemize}
 \item $\RR^3 -\{p\}$, for $p\in\RR^3$ a point.
\item   $\RR^3-\{p_1\cup p_2\}$ where $p_i$ are distinct points?
\item  $\RR^3-\{\ell_1\cup \ell_2\}$ where $\ell_i$ are non-intersecting lines?
\item $\RR^3-\{\ell_1\cup \ell_2\}$, assuming that $l_1$ intersects $l_2$ in exactly one  point?
\end{itemize}
This question is a slightly easier version of the one John Nash asked in class in the movie ``A Beautiful Mind''.  
\end{exercise}

\begin{exercise}[Bonus]
Let $A:S^n\to S^n$ be the antipodal map which sends $x\to -x$.  Begin by showing that this defines an action of $S_2$ on $S^n$ with quotient $S^n/S_2$ diffeomorphic to $\RR P^n$.
\begin{enumerate}
\item Prove that $\Omega^k(S^n)$ decomposes as $\Omega^k_+ \oplus \Omega^k_-$ into $\pm 1$ eigenspaces of the operator $A^*$.
\item Prove that the exterior derivative preserves each of the subspaces $\Omega^\bullet_+$ and $\Omega^\bullet_-$, making each a sub-complex of $(\Omega^\bullet(S^n), d)$.
\item Let $\pi:S^n\to \RR P^n$ be the quotient map.  Show that $\pi^*:\Omega^\bullet(\RR P^n)\to \Omega^\bullet_+$ is an isomorphism of complexes.

\item Use the above results together with knowledge of $H^\bullet (S^n)$ to compute $H^\bullet(\RR P^n)$. 
\end{enumerate}
\end{exercise}




\end{document}
