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On the homotopy groups of a finite dimensional space. (English) Zbl 0538.55010

This paper settles a 30 years old conjecture of 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 \(p^ t\).

MSC:

55Q52 Homotopy groups of special spaces
55Q05 Homotopy groups, general; sets of homotopy classes
55P35 Loop spaces
55P20 Eilenberg-Mac Lane spaces
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology

Citations:

Zbl 0052.195
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