Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
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Najib Idrissi
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Holdsworth88
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Well, it's obvious that the homology groups of a compact cell complex are finitely generated. It's less obvious that compact manifolds can be given CW structures. – Chris Eagle Jul 17 '13 at 12:35
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Following @ChrisEagle's point, see this post (http://mathoverflow.net/questions/36838/are-non-pl-manifolds-cw-complexes) which gives references proving that all compact topological manifolds have the homotopy type of a finite CW-complex (and so in particular have finitely generated (co)homology) – Dan Rust Jul 17 '13 at 13:05
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8The proof is not so bad, here is an outline: Every cpct manifold $X$ embeds into euclidean space (take an open cover $U_i$ and extend each map $U_i \subset \mathbb{R}^{n_i}$ to a map $X \rightarrow S^{n_i}$. Together we get an embedding into a product of spheres, which lives in Euclidean space). One checks that $X\subset \mathbb{R}^N$ is the retract of some open set $U$. Since $X$ is cpct, $X$ is inside some large simplex. Subdivide so that simplices lie entirely in $U$ or $U^c$ and restrict the retraction to those in $U$. Retracts of finite complexes have fg homology, so we're done. – Dylan Wilson Jul 19 '13 at 06:35