Given a smooth affine variety endowed with the Euclidean topology, what sorts of things can one say about topological invariants such a homology and homotopy groups?
There are many beautiful theorems which give very detailed information about topological invariants of projective varieties.
In contrast, when it comes to affine varieties, the only result I know of is that the homology of an affine variety of dimension $n$ is concentrated in degree at most n. Since these are thoroughly studied objects, I assume that one can often say a lot more.
I would be particularly grateful for some references and/or examples.