On the homotopy groups of a finite dimensional space
This paper settles a 30 years old conjecture of \textit{J. P. Serre} in the affirmative. The following result is proved. Theorem: Let X be a 1- connected space and p a prime 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, \(\pi_ nX\) contains a subgroup of order p. Serre conjectured such a result for \(p=2\) in his paper in ibid. 27, 198-232 (1953; Zbl 0052.195). The key ingredient in our proof is the recent powerful result of Haynes Miller: Let X and p be as above. Let \(B=BZ/p\), the classifying space for the group Z/p. Then the space of pointed maps from B to X is weakly contractible; that is, \(\pi_ n(map_*(B,X))=0\) for all \(n\geq 0\). Miller's theorem implies a remarkable property of the iterated loop spaces \(\Omega^ nX\), of such a space X. His result implies that every map from B to \(\Omega^ nX\) is homotopic to the constant map. The proof of our theorem relies on this fact together with some elementary arguments using Postnikov towers. We now conjecture that the following related result is true; given X and p as above, and \(t>1\), if \(p^ t\)-torsion occurs at least once in \(\pi_*X\), then for infinitely many n, \(\pi_ nX\) contains an element of order \(p^ t\).
maps from classifying spaces of cyclic groups into finite dimensional spaces
scientific article; zbMATH DE number 3856005