On the homotopy groups of a finite dimensional space
type: Item
instance of: scholarly article
title: On the homotopy groups of a finite dimensional space
zbMATH Open document ID: 0538.55010
DOI: 10.1007/BF02566349
author: C. A. McGibbon
author: Joseph Neisendorfer
published in: Commentarii Mathematici Helvetici
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139976
review text: 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 \(p^ t\).
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55P20
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 3856005
zbMATH Keywords: Eilenberg-MacLane space
zbMATH Keywords: p-torsion in homology groups
zbMATH Keywords: mod p finite dimensional spaces
zbMATH Keywords: maps from classifying spaces of cyclic groups into finite dimensional spaces
zbMATH Keywords: iterated loop spaces
zbMATH Keywords: Postnikov towers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3856005