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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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.