TITLE: Poincaré Lemma Contractible Hypothesis
QUESTION [8 upvotes]: Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is exact.
What I am wondering is if there is an easy example of a closed differential form on a simply connected domain which is not exact.

REPLY [2 votes]: Because specific examples have already been given, I will make some general statements.
I will assume (without proof) that an $n$ dimensional manifold $M$ is orientable if and only if there is a nonvanishing $n$ form on $M$.
Let $M$ be a compact, orientable, $n$ dimensional manifold without boundary. Then, it is a simple application of Stokes' theorem to show that the nonvanishing form is closed but not exact. To answer your question, we just need to find a simply connected manifold with these properties, for example, $S^n$ for $n \geq 2$.
For an even more general theorem (which is much harder to prove), if $M$ is a connected $n$ dimensional manifold, $H^n_{de}(M) = 0$ if $M$ is non-orientable or non-compact and $H^n_{de}(M) = \mathbb{R}$ otherwise (if it is both orientable and compact.)