De Rham Cohomology Question

Question

Let $M$ be a compact connected $n$-manifold without boundary. Let $\mu\in\Omega^{n-1}(M)$, show that there exists a point $p\in M$ such that $d\mu(p)=0$.

Answer 1

First, assume that $M$ is oriented. Then, by Stokes's theorem, we have $\int_M d\mu = \int_{\partial M} \mu = 0$ since $\partial M = \emptyset$. This can't happen if $d\mu$ is never $0$.

If $M$ is nonorientable, let $N$ by any orientable cover of $M$ (say, the universal cover, or the orientation covering). Let $\pi:N\rightarrow M$ be the covering map.

Then by the previous case, $d(\pi^*\mu)$ is $0$ on some point $p$ of $N$. But $0 = d(\pi^*\mu) = \pi^* (d\mu)$. Since $\pi$ is a local diffeomorphism, $\pi^*$ induces an isomorphism from $\Omega(M)_{\pi(p)}\rightarrow \Omega(N)_{p}$. Since the image of $d\mu$ under this isomorphism is $0$, we must have $d\mu(\pi(p)) = 0$.