Nonabelian algebraic topology
type: Item
instance of: scholarly article
title: Nonabelian algebraic topology
zbMATH Open document ID: 1336.55003
author: Renato Vieira
published in: São Paulo Journal of Mathematical Sciences
publication date: 2015-09-25T00:00:00Z
review text: Connected homotopy \(n\)-types (that is, connected spaces with trivial homotopy groups \(\pi_i X\) for \(i>n\)) admit various algebraic models. More precisely, one wants a suitably algebraic category \(C\) such that some localization of \(C\) is equivalent to the homotopy category of connected homotopy \(n\)-types. For example, connected homotopy \(1\)-types are modeled by groups, and connected homotopy \(2\)-types are modeled by crossed modules, going back to Whitehead. The paper under review is a survey of models for connected homotopy \((n+1)\)-types. The author describes the functors relating crossed \(n\)-cubes of groups, \(n\)-cubes of fibrations, simplicial groups with Moore complex trivial above degree \(n\), and connected \((n+1)\)-types [\textit{J. Loday}, J. Pure Appl. Algebra 24, 179--202 (1982; Zbl 0491.55004), \textit{G. Ellis} and \textit{R. Steiner}, ibid. 46, 117--136 (1987; Zbl 0622.55010), \textit{T. Porter}, Topology 32, No. 1, 5--24 (1993; Zbl 0776.55012)]. He also describes a higher dimensional van Kampen theorem [\textit{R. Brown} and \textit{J. Loday}, ibid. 26, 311--335 (1987; Zbl 0622.55009)]. Section 5 surveys some applications: generalizations of the Blakers-Massey theorem, the homotopy excision theorem, the Hurewicz theorem, and the Hopf formula in group homology, as well as computing certain homotopy colimits of classifying spaces of groups. The paper is expository and contains no new results, except an explicit description of the coproduct of crossed \(n\)-cubes of groups.
reviewed by: Martin Frankland
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 55-02
zbMATH DE Number: 6485539
zbMATH Keywords: homotopy \(n\)-types
zbMATH Keywords: algebraic models
zbMATH Keywords: crossed \(n\)-cubes
zbMATH Keywords: crossed modules
zbMATH Keywords: non-abelian tensor product
zbMATH Keywords: van Kampen
zbMATH Keywords: Blakers-Massey
Wikidata QID: Q56608260
MaRDI profile type: Publication
full work available at URL: s40863-015-0007-3
OpenAlex ID: W2184566472
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DOI: 10.1007/S40863-015-0007-3
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description: scientific article; zbMATH DE number 6485539