Euler characteristic in semialgebraic and other o-minimal groups
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.
groups definable in an o-minimal structure
scientific article; zbMATH DE number 699135