Area preserving pl homeomorphisms and relations in \(K_2\)
type: Item
instance of: scholarly article
title: Area preserving pl homeomorphisms and relations in \(K_2\)
zbMATH Open document ID: 0904.19001
author: Peter Greenberg
publication date: 1998-09-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75273
review text: The author relates the geometry of piecewise-linear homeomorphisms of \(\mathbb{R}^2\) to \(K_2 (\mathbb{R})\) and shows how to use geometric configurations to find nontrivial torsion elements in \(K_2\) of fields. The main theorem states: Let \(\text{SPL}_c\mathbb{R}^2\) be the group of compactly supported, area preserving, piecewise-linear homeomorphisms of the real plane. For each \(g\in\text{SPL}_c\mathbb{R}^2\) there is a function \(Dg: \mathbb{R}^2 \to K_2 (\mathbb{R}) \) such that (1) \(Dg(v)\neq 1\) for only a finite subset of \(v\in \mathbb{R}\); (2) for \(g,h\in\text{SPL}_c\mathbb{R}^2\) we have \(D(gh)(v)=Dg(hv) Dh(v)\); (3) for all \(g\in\text{SPL}_c\mathbb{R}^2 \) we have \(\prod_{v\in\mathbb{R}^2} Dg(v) =1\). With the help of the `three unit formula' of \textit{J. Morita} [|J. Pure. Appl. Algebra 79, No. 1, 51-61 (1992; Zbl 0759.19003)], the map \(Dg\) can be calculated explicitly in terms of Steinberg symbols for many \(g\in\text{SPL}_c\mathbb{R}^2\). Property (3) of the theorem then gives explicit relations among Steinberg symbols and, with a judicious choice of \(g\), allows one to find explicit torsion elements in \(K_2\) of certain finitely generated subfields of \(\mathbb{R}\).
Mathematics Subject Classification ID: 19C20
Mathematics Subject Classification ID: 57Q99
Mathematics Subject Classification ID: 52B45
zbMATH DE Number: 1202020
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: \(K_2\)
zbMATH Keywords: piecewise linear homeomorphisms
zbMATH Keywords: torsion
reviewed by: Kevin Hutchinson
MaRDI profile type: Publication
OpenAlex ID: W2323859781
cites work: Q3972721
cites work: The dilogarithm as a characteristic class for flat bundles
cites work: Geometry of configurations, polylogarithms, and motivic cohomology
cites work: Pseudogroups of \(C^ 1\), piecewise projective homeomorphisms
cites work: Piecewise SL 2 Z Geometry
cites work: Piecewise projective homeomorphisms and a noncommutative Steinberg extension
cites work: Q3828728
cites work: Introduction to Algebraic K-Theory. (AM-72)
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cites work: Third homology of \(\mathrm{SL}(2,\mathbb R)\) made discrete
DOI: 10.5802/AIF.1613
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1202020