Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Let \(M\) be a smooth manifold. What's the definition of an orientation
on \(M\)?

\item[(b)] Suppose \(M\) is a smooth oriented manifold with boundary. What is the
induced orientation on \(\partial M\)?

\item[(c)] State Stokes' Theorem.

\item[(d)] Let \(M\) be a smooth manifold without boundary that is oriented by the
form \(\omega\in\Omega^{\dim(M)}(M)\). Suppose that
\([0,1]=\{x\in\R\mid 0\le x\le 1\}\) is oriented by the form \(dx\), and that
\(M\times[0,1]\) has the orientation defined by \(\omega\wedge dx\). Endow
\(\partial(M\times[0,1])=(M\times\{0\})\cup(M\times\{1\})\) with the orientation
induced by \(M\times[0,1]\) and let \(i_0:M\to M\times\{0\}\) and
\(i_1:M\to M\times\{1\}\) be the obvious diffeomorphisms. Prove that \(i_0\)
preserves orientation if and only if \(i_1\) does not.

\item[(e)] Let \(f:M\to N\) and \(g:M\to N\) be smooth maps between smooth manifolds
without boundaries. What is the definition of \(f\) and \(g\) being smoothly
homotopic to each other?

\item[(f)] Suppose \(M\) is a smooth, compact, oriented manifold without boundary.
Suppose \(N\) is a smooth manifold and that \(f:M\to N\) and \(g:M\to N\) are smoothly
homotopic smooth maps. Prove that if \(\theta\in\Omega^k(N)\) for some
\(k>\dim(M)\), then
\[
  \int_M f^*\theta=\int_M g^*\theta.
\]
\end{enumerate}