Misunderstanding of de Rham Cohomology theorems

Question

There's a theorem in my book that says the nth de Rham cohomology of a contractible space is zero for positive n. Another theorem says that if a manifold is orientable, connected, and compact, then the top de Rham comology has rank 1. I'm asking why these two theorems don't contradict each other. For example, the nth unit ball satisfies the conditions of both theorems, so it has top de Rham cohomology group equal to both 1 and 0. Thanks for helping me clear up the misunderstanding.

Answer 1

The ball is not a "manifold" in the sense of the theorem you mentioned. If the ball is open then it is not compact. If the ball is closed then it has boundary so it is not a "manifold".