Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

I would like to determine the homotopy type of a torus with 8 punctures.

(I have come across this problem studying deformations of discontinuous groups of Heisenberg groups...)

Other than trying really hard to visualize, are there any other methods for finding homotopy types of punctured surfaces or the like?

share|cite|improve this question
up vote 5 down vote accepted

A torus with one puncture is homotopy equivalent to the wedge of two circles, which is a $K(F_2, 1)$. To see this, think of the torus as a square with sides identified. Puncturing the square allows you to retract onto the $1$-skeleton.

It follows that a torus with $n+1$ punctures is homotopy equivalent to the wedge of $n+2$ circles, which is a $K(F_{n+2}, 1)$. To see this, start with the wedge of two circles and thicken some part of it back up to $2$ dimensions, then puncture it.

share|cite|improve this answer
Puncture, retract, puncture, retract... Thanks for the quick answer. – Earthliŋ Jun 17 '12 at 3:21

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.