Can a non-compact manifold have infinite-dimensional cohomology?
For compact manifolds, Hodge Theory tells us that (de Rham) cohomology is finite dimensional. What about non-compact manifolds? That is:
Can non-compact manifolds have infinite dimensional cohomology?
If the answer is yes, is there an example for which this is easy to see?
The (non-compact) manifold $\mathbb R^2\setminus \mathbb Z^2$ should have infinite first homology and cohomology.
Sure. Here are two examples:
Let $M$ be the real line with all integer points removed. Then $M$ is the disjoint union of countably many open intervals, and $H^0 (M, \mathbb{R})$ is a real vector space of uncountably infinite dimension. (To be precise, it is the dual of the real vector space of countably infinite dimension.)
Let $M$ be the real plane with the set $\{ (n, 0) : n \in \mathbb{Z} \}$ removed. It's not too hard to see that $M$ is homotopy-equivalent to a chain of countably many circles joined side-by-side. It follows that $H^1 (M, \mathbb{R})$ is a real vector space of uncountably infinite dimension – we can concoct differential forms with a prescribed singularity at each hole independently, and their cohomology classes are distinct because they can be distinguished by integrating along a loop around the relevant singularities.