Nerves, fibers and homotopy groups
type: Item
instance of: scholarly article
title: Nerves, fibers and homotopy groups
zbMATH Open document ID: 1030.55006
DOI: 10.1016/S0097-3165(03)00015-3
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-05-27T00:00:00Z
review text: A much used result of \textit{D. Quillen} [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)] is that a map of posets \(f:P\to Q\) with contractible fiber is a homotopy equivalence (simplicial complexes are associated to posets in the usual way). The first main result of the paper under review extends this by only requiring the map to be \(k\)-connected and concluding that \(f\) induces an isomorphism on homotopy groups through dimension \(k\). The second result is a similar extension of \textit{K. Borsuk}'s theorem on nerves [Fundam. Math. 35, 217-234 (1948; Zbl 0032.12303)] that a simplicial complex \(P\) is homotopy equivalent to the nerve of any cover of \(P\) by subcomplexes with all intersections either empty or contractible. Namely, if the condition on intersections is weakened to connectivity through a range (depending on the number of subcomplexes that are being intersected), then there is a map from \(P\) to the nerve of the cover that induces an isomorphism on homotopy groups through a range. (Quillen's and Borsuk's results follow from these extensions by the Whitehead theorem.).
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 1917909
zbMATH Keywords: poset
zbMATH Keywords: Borsuk's nerve theorem
zbMATH Keywords: Quillen's fiber theorem
author: Anders Bjoerner
MaRDI profile type: Publication
cites work: Q4871782
cites work: Chessboard Complexes and Matching Complexes
cites work: On the imbedding of systems of compacta in simplicial complexes
cites work: Q3136738
cites work: Q5607712
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
cites work: Homotopy type and Euler characteristic of partially ordered sets
Recommended article: Fiber Bundles and Homotopy
Recommended article: A variation on the homological nerve theorem
Recommended article: Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers
Recommended article: Principal bundle groupoids, their gauge group and their nerve
Recommended article: Fibrewise homology
Recommended article: Homologous fibres and total spaces
Recommended article: Q3335569
Recommended article: Higher nerves of simplicial complexes
Recommended article: Classifying space via homotopy coherent nerve
Recommended article: The structural nature of the nerve functor for \(n\)-groupoids
description: scientific article; zbMATH DE number 1917909