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

Can somebody explain why the fundamental group of a connected compact n-manifold M is finitely generated? I know that this manifold is homotopic to a CW complex (and I guess connected, because M is connected). Now what is the relation between the fundamental group and cell decomposition? (I need complete details)

share|cite|improve this question

If the manifold is compact, the CW-complex that it is homotopy-equivalent to is also compact. Compact CW-complexes have a finite number of cells. Feed in the standard process to generate a presentation of $\pi_1$ of a CW-complex and that's the argument.

share|cite|improve this answer
Thanks Ryan. As I said my problem is that I don't know how to describe the fundamental group in terms of cell decomposition of the CW complex. – mandegar Dec 8 '11 at 7:51
It's an application of van Kampen's theorem, which will probably be in many textbooks of algebraic topology. See for example, prop 1.26 on p.50 of Hatcher. – Soarer Dec 8 '11 at 8:01
Ryan, I think I got it: do you mean that the number of generators of \pi_1 is the same as the number of 1-cells? and relators come from 2-cells I guess? – mandegar Dec 8 '11 at 8:02
The number of generators is the number of 1+cells minus the number of 0-cells, plus 1. You have to collapse a maximal tree in the 1-skeleton to get a presentation unless you're willing to work with groupoids. Relators come from 2-cells, yes. – Ryan Budney Dec 8 '11 at 15:41
Thanks a lot Ryan and Soarer. I got it. – mandegar Dec 8 '11 at 19:32

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.