2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58953Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups. The term "cells" might be misleading because their topology can be quite non-trivial. As a first step towards understanding this topology, we enumerate the connected components of real double Bruhat cells. This result extends (from the simply-laced case to the general one) and proves the conjecture made in a joint work with B.Shapiro-M.Shapiro-A.Vainshtein; it also extends earlier work by B.Shapiro-M.Shapiro-A.Vainshtein and K.Rietsch.18 pages, 4 figures; the title corrected, the abstract slightly modified, one reference added, the proof of Lemma 4.4 simplifiedAlgebraic GeometryRepresentation Theorytext