Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It's known that there exists a deformation retraction from the space $\mathbb{R}^3$\ $S^1$ to $S^2 \wedge S^1$, and I thought I had a visualization for it, but now it seems discontinuous. Can anyone help out with describing (or even better constructing explicitly) this map?


share|cite|improve this question
up vote 11 down vote accepted

The space $\mathbb R^3\setminus S^1$ is obtained by rotating a closed half-plane with a puncture in it. The latter space can be retracted onto semi-circle plus a diameter. The rotation of semi-circle plus a diameter creates $S^2$ plus a diameter. Now it is time to lose the rotational symmetry: we contract the Eastern hemisphere into a point (sorry, guys). The result is $S^2\wedge S^1$.

share|cite|improve this answer
+1 Very nice , Leonid ! (Especially since I live in the Western hemisphere...) – Georges Elencwajg Jun 26 '12 at 19:55
perfect, I can see it now! Thanks so much! – MGN Jun 26 '12 at 21:25

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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