The cohomology of homotopy categories and the general linear group
type: Item
instance of: scholarly article
title: The cohomology of homotopy categories and the general linear group
zbMATH Open document ID: 0701.18009
DOI: 10.1007/BF00584524
author: Hans-Joachim Baues
author: Winfried Dreckmann
published in: \(K\)-Theory
publication date: 1989-01-01T00:00:00Z
review text: The authors define cohomology groups of a very general kind for small categories and consider their properties. The second cohomology groups classify certain kinds of extensions. If \({\mathbb{C}}\) is a suitable subcategory of the homotopy category, then it has a canonical 3- dimensional cohomology class \(<{\mathbb{C}}>\), which acts as a universal Toda bracket. Several examples are given. In particular, there is an elegant description of Igusa's associativity class \(\chi\) (1) in the cohomology of \(GL_ n({\mathbb{Z}})\). It is got by considering wedges of spheres and automorphisms of nil(2)-groups (groups of the form F/[F,[F,F]] with F a free group).
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 55Q35
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 19D10
zbMATH DE Number: 4148383
zbMATH Keywords: cohomology of categories
zbMATH Keywords: secondary homotopy operations
zbMATH Keywords: cohomology groups
zbMATH Keywords: small categories
zbMATH Keywords: second cohomology groups
zbMATH Keywords: homotopy category
zbMATH Keywords: universal Toda bracket
zbMATH Keywords: Igusa's associativity class
zbMATH Keywords: nil(2)-groups
reviewed by: Richard John Steiner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00584524
OpenAlex ID: W1966758824
description: scientific article; zbMATH DE number 4148383