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\newcommand{\myTitle}{HW 7, due Friday, March 29}
\newcommand{\myClass}{Math 411}
\newcommand{\myName}{My name goes here}

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\begin{document}

\begin{center}
{\large \myClass}\\[5pt]
\myTitle
\smallskip

\myName
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% main body starts here
\begin{enumerate}
\item Explain why the components and the path components of a manifold
  are the same. (Quote the right result from the handout, and argue that it
  applies.)
  \medskip

  {\sc solution:}

\item Let $X$ be a topological space.
\begin{enumerate}
\item Suppose $X$ is not connected.  Show that there exist nonempty,
  disjoint subsets $A$ and $B$ of $X$ such that both $A$ and $B$ are
  open and $X=A\cup B$.
  \medskip

  {\sc solution:}

\item Let $Y\subset X$ with the subspace topology.  Suppose $X$ is not
  connected, and write $X=A\cup B$, as in the previous exercise.  If
  $Y$ is connected, show that $Y$ is contained in either $A$ or $B$.
  \medskip

  {\sc solution:}

\end{enumerate}

\item Let $X$ and $Y$ be topological spaces, and let $f\colon X\to Y$ be any
  mapping (of sets).  We say $f$ is {\em locally constant} if for
  each $x\in X$ there exists an open neighborhood $U$ of $x$ such that
  $f$ restricted to $U$ is constant.
  \begin{enumerate}
\item Prove that if $f$ is locally constant, it is continuous.
  \medskip

  {\sc solution:}

\item Prove that if $f$ is locally constant, then it is constant on
  each connected component of $X$.
  \medskip

  {\sc solution:}

\item Let $M$ be a manifold.  Prove that $H^0M\approx\R^c$ where $c$
  is the number of connected components of $M$.
  \medskip

  {\sc solution:}

\end{enumerate}
\item Let $f\colon M\to N$ be a mapping of manifolds.  
  \begin{enumerate}
\item Prove that for
  each $k\geq0$, the pullback $f^*\colon\Omega^kN\to\Omega^kM$ induces
  a well-defined mapping $f^*\colon H^kN\to H^kM$.
  \medskip

  {\sc solution:}

\item Prove that if $f$ is constant, then $f^*\colon H^kN\to H^kM$ is
  the zero map if $k>0$.
  \medskip

  {\sc solution:}

\item Prove that if $f$ is constant and both $M$ and $N$ are
  connected, then $f^*\colon H^0N\to H^0 M$ is an isomorphism.  (Note: $H^0M\approx H^0N\approx\R$ in this case.)
  \medskip

  {\sc solution:}

\end{enumerate}
\item Suppose $M$ is a contractible manifold and $h$ is a
  homotopy between the identity mapping on $M$ and a constant mapping.  Let
  $\omega\in\Omega^kM$ be a cocycle, i.e., $d\omega=0$.  We saw in class that if
  $k\geq1$, then $\omega=dP(h^*\omega)$ where $P$ is the prism operator.

  Let $F$ be vector field on $\R^3$ with $\mbox{curl}\,F=0$.  
    Calculate $\phi\colon\R^3\to\R$ such that $\mbox{grad}\,\phi=F$ by
      calculating $P(h^*\omega)$ where $\omega$ is a suitably defined $1$-form
      on $\R^3$ and $h(x,y,z)=(tx,ty,tz)$ for $t\in[0,1]$ is a homotopy of the
      zero-mapping and the identity on $\R^3$.
    Show that for each $p\in\R^3$,
      \[
      \phi(p)=\int_{\gamma}F\cdot d\vec{t},
      \]
      the flow of $F$ along $\gamma$ where $\gamma(t):=tp$ for $t\in[0,1]$.
  \medskip

  {\sc solution:}

\item A {\em Lie group} is a manifold $G$ with a group structure so that both
  the multiplication and inverse mappings:
  \begin{align*}
    G\times G&\to G\\
    (g,h)&\mapsto gh\\[10pt]
    G&\to G\\
    g&\mapsto g^{-1}
  \end{align*}
  are smooth.
Let $G$ be a connected Lie group, and let $U$ be an open neighborhood of the
identity.  Show that $U$ generates $G$.  (Define $U^n=\{g_1\cdots g_n:
g_1,\dots,g_n\in U\}$. You must show that $\cup_{n\geq 1}U^n=G$.)
  \medskip

  {\sc solution:}

\end{enumerate}
\end{document}
