 Serre finiteness theorem for homotopy groups

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.