2020/11/07 by Jiming Ma, Zixi Wang, Ma, Jiming +1
Computer Science · Mathematics · #20E18 #57M05 #57N16 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2011.03784
openalex publication_date 2020/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is ℍ4, ℍ2ℂ or ℍ2 × ℍ2. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold M, we show that whether M is geometric could be detected by the profinite completion of its fundamental group.