Math 6510 • Differentiable Manifolds

\begin{enumerate}

\item
Construct an explicit \(n\)-form \(\omega\) on \(\mathbb{R}^n\) with compact support such
that \(\omega\) is closed but there does not exists an \((n-1)\)-form \(\eta\) with
compact support and \(d\eta=\omega\). You can use the existence of bump
functions in your construction but you need to prove that your
construction works.


\item
Let \(M\) be a smooth, compact orientable \(n\)-manifold without bound-
ary and let \((U,\phi)\) be a chart for \(M\). Show that there exists a
closed \(n\)-form on \(M\) with support in \(U\) that is not exact.

\end{enumerate}