I think this is a simple question, I hope it is not too soft. I used to think that the smaller the cohomology of an object is, the better it is behaved. For instance, the degree to which the existence and uniqueness of a global primitive of differentials forms on a smooth manifold that have local primitives fail is controlled by the (de Rham) cohomology, the failure of an exact sequence $0 \to A \to K \to G \to 0$ of groups to be split is controlled (under suitable hypotheses) by the group cohomology $H(G,A)$, etc. And more generally there are many invariants whose non-triviality is correlated (as far as I know) both to the cohomology being non-trivial and the object being complicated, like the Galois group, the fundamental group or the $K$-theory rings.
Isn't it paradoxical then that orientability, which seems to me a good property to have, is equivalent (for compact topological manifolds without boundary) to having non-zero top-degree (singular integral) cohomology?