See Hatcher p.146-147 theorem 2.44: http://www.math.cornell.edu/~hatcher/AT/ATch2.pdf(adsbygoogle = window.adsbygoogle || []).push({});

In the proof, he puts himself in the more general situation where he has a bounded chain complex of finitely generated groups and goes on to prove that the alternating sum of the ranks of these groups equals the alternating sum of the ranks of their homology groups. From there, the result follows as a special case (in view of lemma 2.34 and theorem 2.35).

The proof relies of the little algebraic fact stated between the statement of the theorem and its proof. I believe that in this little algebraic fact (which is trivial given the fundamental theorem on finitely generated abelian groups), the condition that the groups be finitely generated is important.

What I wonder is, in the proof, why are the cycle groups Z_n=Ker(d_n) finitely generated??

(Note that in the case that interests us for the statement of the theorem this is immediate because the groups in the cellular chain complex are actually finitely generatedfreeabelian and subgroups of finitely generated free abelian groups are themselves finitely generated free abelian...)

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# Question of algebraic flavor in algebraic topolgy

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