Gottlieb groups of spheres
type: Item
instance of: scholarly article
title: Gottlieb groups of spheres
zbMATH Open document ID: 1172.55004
author: Juno Mukai
author: Marek Golasiński
published in: Topology
publication date: 2009-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0408256
review text: Let \(X\) be a connected space. The \(k\)-th Gottlieb group \(G_{k}(X)\) of \(X\) is the subgroup of the \(k\)-th homotopy group \(\pi_{k}(X)\) consisting of all elements which can be represented by a map \(f : S^{k} \rightarrow X\) such that \(f \vee id_{X}: S^{k} \vee X \rightarrow X\) extends (up to homotopy) to a map \(F:S^{k} \times X \rightarrow X\). This paper takes up a systematic study of the Gottlieb groups of spheres, \(G_{n+k}(S^{n})\), for \(k \leq 13\) using classical homotopy theoretic methods. The paper determines these groups except for the \(2\)-primary components in the cases: \(k=9, n= 53; k=11, n=115\). The main tools of the paper are the composition methods developed by Toda in his classic study of the homotopy groups of spheres, and builds on the work of Hoo, Mahowald, and Kervaire. The results are compiled into a series of tables for \(k=1\) up to \(k=13\) and \(n=2\) up to \(n=26\).
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 5597576
zbMATH Keywords: EHP sequence
zbMATH Keywords: fibration
zbMATH Keywords: Gottlieb group
zbMATH Keywords: rotation group
zbMATH Keywords: Stiefel manifold
zbMATH Keywords: Toda bracket
zbMATH Keywords: Whitehead product
Wikidata QID: Q105799578
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
OpenAlex ID: W4206322086
arXiv ID: math/0408256
DOI: 10.1016/J.TOP.2007.11.001
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description: scientific article