Topology of definable abelian groups in o-minimal structures
Let \(M\) be an o-minimal expansion of a real closed field. For every positive integer \(n\) the \(n\)-torus over \(M\), \(T_n(M)\), is the poly-interval \([0, 1)^n\) in \(M\) with the sum operation modulo \(1\). Let \(G\) be a definably connected, definably compact definable abelian group of dimension \(n\) in \(M\), equipped with the Pillay topology. \textit{A. Berarducci, M. Mamino} and \textit{M. Otero} showed [Isr. J. Math. 180, 143--161 (2010; Zbl 1213.03048), Theorem 3.4] that \(G\) is definably homotopy equivalent to \(T_n(M)\). On the other hand it is not true in general that \(G\) is also definably isomorphic to \(T_n(M)\). Here the authors deal with the question whether \(G\) is definably homeomorphic to \(T_n(M)\). A positive answer is given when \(n \neq 4\). Moreover it is proved that, if \(G\) is semialgebraic, namely \(M\) is a real closed field without any additional structure, then the result is true in all dimensions.
scientific article; zbMATH DE number 6048032