Some Good Algebraic Topology Exercises

I am teaching a topology prep course for first year graduate students taking their qualifying exams. I have been able to think of about ten days' worth of exercises, but am running out of ideas. Does anyone have any good questions or a place to find them? I am looking for exercises involving singular homology that are not just "Compute homology of" type questions. In particular I need some good Euler characteristic, degree of mapping, and Jordan-Alexander Complement problems. Though, any questions at all are surely welcome. The class they took is equivalent to Chapter 2 of Hatcher's 'Algebraic Topology' book.

The book by Tammo Tom Dieck has a wealth of good non-computational exercises.

I like questions of the sort:

1) Suppose $p: \mathbb{R}P^2 \to X$ is a covering map. Prove that $p$ is a homeomorphism.

2) Suppose $p: \mathbb{C}P^2 \to X$ is a covering and suppose that $X$ is a manifold. Again show that $p$ is a homeomorphism.

(the first one works only by looking at the euler characteristic, for the second one I guess one needs Poincaré duality)..

3) Show that there is no map $f: S^n\to S^1$ which is $\mathbb{Z}/2\mathbb{Z}$ equivariant (that is $f(-x) = -f(x)$).