Area preserving pl homeomorphisms and relations in \(K_2\)
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 \(\mathbb{R}\).
scientific article; zbMATH DE number 1202020
scientific article