Combinatorial group theory and the homotopy groups of finite complexes
type: Item
instance of: scholarly article
title: Combinatorial group theory and the homotopy groups of finite complexes
zbMATH Open document ID: 1270.55011
author: Roman Mikhailov
author: Jie Wu
published in: Geometry \& Topology
publication date: 2013-04-09T00:00:00Z
review text: The calculation of homotopy groups of spheres and Moore spaces is a key problem in algebraic topology. \textit{J. Wu} in the paper [Math. Proc. Camb. Philos. Soc. 130, No. 3, 489--513 (2001; Zbl 0986.55013)] obtains a combinatorial description of the homotopy groups of \(S^2\). \textit{F. R. Cohen} and \textit{J. Wu} [Q. J. Math. 62, No. 4, 891--921 (2011; Zbl 1252.55005)] continue this analysis in a way which leads to the present paper. Using Milnor's FK construction, braid groups, free products with amalgamation, simplicial theory, etc., the authors obtain remarkable descriptions of the homotopy groups of spheres as the centers of explicitly constructed groups. They also obtain such descriptions of \(\pi_n\) of Moore spaces when \(n\neq 3\). As one might expect, the paper uses a considerable amount of combinatorial group theory. As a minor quibble, the reviewer would expect a reference (out of a total of 26) to [\textit{E. H. Brown jun.}, Ann. Math. (2) 65, 1-20 (1957; Zbl 0077.16804)].
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 6152262
zbMATH Keywords: homotopy groups
zbMATH Keywords: braid groups
zbMATH Keywords: free product with amalgamation
zbMATH Keywords: simplicial groups
zbMATH Keywords: spheres
zbMATH Keywords: Moore spaces
zbMATH Keywords: Brunnian words
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: gt.2013.17.235
OpenAlex ID: W2115399592
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DOI: 10.2140/GT.2013.17.235
description: scientific article; zbMATH DE number 6152262