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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)

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1 Answer 1

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.

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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
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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
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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

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