[Math] Some Good Algebraic Topology Exercises

Question

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.

Answer 1

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