The Euler class in homological algebra
type: Item
instance of: scholarly article
title: The Euler class in homological algebra
zbMATH Open document ID: 1229.55012
published in: Journal of Pure and Applied Algebra
publication date: 2011-10-21T00:00:00Z
review text: The topological group \(G_\tau\) of orientation preserving homeomorphisms of the circle is homotopy equivalent to \(S^1 = U(1)\). It is not abelian however, but simple, and the identity \(G_\delta \to G_\tau\) from the same group equipped with the discrete topology induces a homology equivalence \(BG_\delta \to BG_\tau \simeq \mathbb C P^\infty\). The author provides a computational proof of this fact (due to Mather and generalized by Thurston) based on bisimplicial models. He also constructs a group extension \(0 \to \mathbb Z \to G_{-} \to G_\delta \to 1\), which he calls ``the discrete Euler Class'' of \(G\) since it corresponds via the above homology equivalence to the usual, topological, Euler Class. The value of this Euler Class is shown to be bounded on any space.
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 5960892
zbMATH Keywords: group of homeomorphisms
zbMATH Keywords: Euler class
zbMATH Keywords: group extension
zbMATH Keywords: Friedlander-Milnor conjecture
author: Solomon Jekel
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.03.005
OpenAlex ID: W2034206724
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DOI: 10.1016/J.JPAA.2011.03.005
description: scientific article; zbMATH DE number 5960892