Euler characteristic in semialgebraic and other o-minimal groups
type: Item
instance of: scholarly article
title: Euler characteristic in semialgebraic and other o-minimal groups
zbMATH Open document ID: 0815.03024
DOI: 10.1016/0022-4049(94)90127-9
published in: Journal of Pure and Applied Algebra
publication date: 1994-12-01T00:00:00Z
review text: This is a very interesting paper applying Euler characteristic to groups definable in an o-minimal structure. The setting is an o-minimal structure \(\mathcal R\) with the definable choice property (a strengthening of elimination of imaginaries which holds, for example, if the domain of \(\mathcal R\) has a definable group structure). There is a well-defined notion of Euler characteristic: if \(C\) is a cell then its Euler characteristic \(E(C)= (-1)^{\dim(C)}\), and if \(X\) is a definable set then \(E(X)= \sum E(C)\), a sum over cells in some cell-decomposition of \(X\). If \(p\) is a prime or 0, then a \(p\)-group is a definable group \(G\) such that for every proper definable subgroup \(H\) of \(G\), \(E(G/H)\equiv 0\text{ mod }p\) (where `\(\equiv\text{mod }0\)' means `\(=\)'). Analogues of the familiar Sylow theorems are proved, and there are also results showing, for example, that the set of all Sylow 0-subgroups of a definable group \(G\) is parametrizable (in a technical sense) with Euler characteristic 1. Among many other interesting results, it is shown that if \(G\) is a definable group with \(E(G)= \pm1\) for some \(n\geq 1\) then the map \(x\mapsto x^ n\) is a bijection on \(G\), and that if \(G\) is a definable Abelian group with \(E(G)\neq 0\) then there are definable \(H,T\leq G\) with \(G= H\times T\), \(E(H)= \pm1\) and \(T\) finite. A notion of torus is introduced, and maximal tori in a definable group are shown to be conjugate. It is shown that any definable group of finite exponent is finite. There is discussion of several semi-algebraic examples.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 699135
zbMATH Keywords: semialgebraic group
zbMATH Keywords: Sylow subgroup
zbMATH Keywords: Euler characteristic
zbMATH Keywords: groups definable in an o-minimal structure
zbMATH Keywords: definable group
author: Adam Strzeboński
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90127-9
OpenAlex ID: W2089355513
cites work: A generalization of the Tarski-Seidenberg theorem, and some nondefinability results
cites work: Q4044675
cites work: Definable Sets in Ordered Structures. II
cites work: On groups and fields definable in o-minimal structures
cites work: Definable Sets in Ordered Structures. I
cites work: Q3978988
Wikidata QID: Q127613916
description: scientific article; zbMATH DE number 699135