On the homotopy groups of finite dimensional CW-complexes (Q582625)

Let p be a prime and \({\mathbb{Z}}_ p\) the cyclic group of order p. \textit{C. A. McGibbon} and \textit{J. A. Neisendorfer} have proved the following result [Comment. Math. Helv. 59, 253-257 (1984; Zbl 0538.55010)]: Let X be a 1-connected CW-complex such that (i) \(H_ n(X;Z_ p)\neq 0\) for some \(n>0\), and (ii) \(H_ n(X;Z_ p)=0\) for all n sufficiently large. Then for infinitely many n, \(\pi_ n(X)\) contains a subgroup of order p. In this letter the author announces without proof extensions of this result. An example is Theorem C. Let X be a nilpotent connected CW- complex such that (i) \(\pi_ 1(X)\) contains a subgroup of order p, and (ii) \(H_ n(X;Z_ p)=0\) for all n sufficiently large. Then for infinitely many n, \(\pi_ n(X)\) contains a subgroup of order p.