2006/04/09 by Alessandro Berarducci, Berarducci, Alessandro
Mathematics · #03C64 #03H05 #22E15 #Advanced Topics in Algebra #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #math.GR #math.LO #msc:03C64 #msc:03H05 #msc:22E15
paper · pdf · doi:10.48550/arxiv.math/0604186
21 pages
arxiv created 2006/04/09 · openalex publication_date 2006/04/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal ``infinitesimal subgroup'' G00 such that the quotient G/G00, equipped with the ``logic topology'', is a compact (real) Lie group. Our first result is that the functor G↦ G/G00 sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum \widetilde G of G. We prove that G/G00 is a topological quotient of \widetilde G. We thus obtain a natural homomorphism Ψ^* from the cohomology of G/G00 to the (Čech-)cohomology of \widetilde G. We show that if G00 satisfies a suitable contractibility conjecture then \widetilde G00 is acyclic in Čech cohomology and Ψ^* is an isomorphism. Finally we prove the conjecture in some special cases.