Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Define the DeRham cohomology groups \(H^k(M)\) for a smooth manifold
\(M\).
\item[(b)] Let \(N\) be a compact, smooth, \(k\)-dimensional manifold without
boundary and \(f:N\to M\) a smooth map. Let \(\alpha\in\Omega^k(M)\) be a closed
\(k\)-form on \(M\) with \(\int_N f^*\alpha\ne0\). Either show that \(\alpha\)
represents a non-zero element of \(H^k(M)\) or find a counterexample.
\item[(c)] The wedge product defines a map
\(\Omega^k(M)\times\Omega^l(M):\Omega^{k+l}(M)\) by
\((\alpha,\beta)\mapsto\alpha\wedge\beta\). Show that this descends to a map from
\(H^k(M)\times H^l(M)\) to \(H^{k+l}(M)\).
\end{enumerate}