On the relation between the fundamental groupoids of the classifying space and the nerve of an open cover
type: Item
instance of: scholarly article
title: On the relation between the fundamental groupoids of the classifying space and the nerve of an open cover
zbMATH Open document ID: 0583.55014
DOI: 10.1016/0022-4049(85)90088-X
author: Abdul Razak Salleh
author: John Taylor
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: Associated to an open cover U of a topological space X are the classical simplicial nerve of U, KU, and the classifying space of U, BU. There is a natural map BU\(\to KU\) and an induced morphism on fundamental groupoids \(\pi\). In the case that all the elements of the cover are open and path connected the authors show that \(\pi\) BU\(\to \pi KU\) is a quotient morphism and compute the normal subgroupoid in terms of generators and relations. Several relationships are noted between the nature of the morphism \(\pi\) BU\(\to \pi KU\) and work of others concerning actions of groups and transformation groups.
Mathematics Subject Classification ID: 55U05
Mathematics Subject Classification ID: 57S99
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 3934029
zbMATH Keywords: open cover
zbMATH Keywords: simplicial nerve
zbMATH Keywords: classifying space
zbMATH Keywords: fundamental groupoids
zbMATH Keywords: actions of groups
reviewed by: Solomon Jekel
MaRDI profile type: Publication
cites work: Q4113141
cites work: Lifting homotopies through fixed points
cites work: Homotopy invariant algebraic structures on topological spaces
cites work: Q5545247
cites work: Groupoids as Coefficients
cites work: The fundamental groupoid as a topological groupoid
cites work: On the Connection between the Second Relative Homotopy Groups of Some Related Spaces
cites work: Colimit theorems for relative homotopy groups
cites work: A van Kampen theorem for unions of non-connected spaces
cites work: Q5614474
cites work: Q4076522
cites work: Q4679162
cites work: Q3672723
cites work: Groups of homeomorphisms of a simply connected space
cites work: ČEch Homotopy I
cites work: Cech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory
cites work: On the Fundamental Group of a Transformation Group
cites work: On Lifting Transformation Groups
cites work: Classifying spaces and spectral sequences
cites work: Q3341041
cites work: Q5674144
cites work: Q5522742
cites work: Lectures on algebraic and differential topology
cites work: Generators and relations for certain special linear groups
description: scientific article; zbMATH DE number 3934029