Note on the Serre conjecture theorem of McGibbon and Neisendorfer
type: Item
instance of: scholarly article
title: Note on the Serre conjecture theorem of McGibbon and Neisendorfer
zbMATH Open document ID: 0807.55010
DOI: 10.1016/0166-8641(94)90118-X
published in: Topology and its Applications
publication date: 1995-03-01T00:00:00Z
review text: Let \(X\) be a space so that it has a nontrivial, finitely generated, \(Z_ p\) homology. If \(X\) is simply connected, Neisendorfer and McGibbon have shown that infinitely many homotopy groups contain a subgroup of order \(p\). (This is Serre's conjecture.) If \(X\) is not simply connected, the conclusions do not hold. The author investigates conditions on \(\pi_ 1 (X)\) so that the conclusions still hold. They center around various concepts of nilpotence. As an application, the following interesting corollary is proved: Any connected finite \(H\)-space \(X\) has the homotopy type of a point or a torus or else \(\pi_ n (X)\) contains a subgroup of order \(p\) for infinitely many primes.
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 500695
zbMATH Keywords: \(p\)-nilpotent spaces
zbMATH Keywords: fundamental group
zbMATH Keywords: nontrivial finitely generated \(Z_ p\) homology
zbMATH Keywords: subgroup of order \(p\)
zbMATH Keywords: connected finite \(H\)-space
zbMATH Keywords: homotopy groups
Wikidata QID: Q123346061
author: Wenhuai Shen
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 500695