Torsion in homotopy groups of CW complexes (Q1202829)

The \(p\)-torsion in homotopy groups \(\pi_ n X\) of a CW-complex \(X\) is studied. The main results are the following theorems: Theorem 4. Let \(X\) be a connected \(p\)-nilpotent CW-complex. If \(X\) satisfies one of the following conditions: Condition 1. (i) \(\pi_ 1 X@>p>> \pi_ 1 X\) is not injective, and (ii) \(H_ n(X;Z_ p)=0\) for all \(n\) sufficiently large. Condition 2. (i) \(\pi_ 1 X@>p>> \pi_ 1 X\) is bijective, (ii) \(H_ n(X;Z_ p)\neq 0\) for some \(n>0\), and (iii) \(H_ n(X;Z_ p)=0\) for all \(n\) sufficiently large; then for infinitely many \(n\), \(\text{Tor}(\pi_ n X,Z_ p)\neq 0\), where \(\pi_ 1 X@>p>> \pi_ 1X\) is defined by \(p(a)=a^ p\). Theorem 5. Let \(X\) be a connected finite dimensional CW-complex with nilpotent fundamental group such that (i) \(\pi_ 1 X@>p>>\pi_ 1 X\) is bijective, and (ii) \(H_ n(X,Z_ p)\neq 0\) for some \(n>0\). Then for infinitely many \(n\), \(\text{Tor}(\pi_ n X,Z_ p)\neq 0\).