= Serre finiteness theorem for homotopy groups
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A simply connected <CW complex> of finite type has finitely generated <homotopy groups> in every degree. More generally, finite generation of its integral <homology> groups in each degree suffices. This is a finiteness theorem, not a claim that the groups are finite. A finite <fundamental group> allows the same conclusion after passing to a <universal cover> with finite type: a finite number of base cells lifts to a finite number of covering cells in each degree. It permits higher <homotopy> to be killed with finitely many cell attachments per dimension.
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