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)
Fundamental group of connected compact n-manifolds
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algebraic-topology
1 Answers
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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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0Thanks 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. – 2011-12-08
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3It'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. – 2011-12-08
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0Ryan, 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? – 2011-12-08
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1The 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. – 2011-12-08
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0Thanks a lot Ryan and Soarer. I got it. – 2011-12-08