10
$\begingroup$

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.

$\endgroup$
  • 3
    $\begingroup$ What about questions from past qualifying exams? $\endgroup$ – wildildildlife Jul 17 '12 at 14:49
  • $\begingroup$ @wildildildlife I've been doing the rounds, looking up old exams online. Most of what I see I've already done or doesn't apply. $\endgroup$ – Joe Johnson 126 Jul 17 '12 at 16:26
3
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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 $f(-x) = -f(x)$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.