2023/02/23 by Fernando Figueroa, Figueroa, Fernando, Joaquín Moraga +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2302.11790
openalex publication_date 2023/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we study the fundamental groups of low-dimensional log canonical singularities, i.e., log canonical singularities of dimension at most 4. In dimension 2, we show that the fundamental group of an lc singularity is a finite extension of a solvable group of length at most 2. In dimension 3, we show that every surface group appears as the fundamental group of a 3-fold log canonical singularity. In contrast, we show that for r≥ 2 the free group Fr is not the fundamental group of a 3-dimensional lc singularity. In dimension 4, we show that the fundamental group of any 3-manifold smoothly embedded in ℝ4 is the fundamental group of an lc singularity. In particular, every free group is the fundamental group of a log canonical singularity of dimension 4. In order to prove the existence results, we introduce and study a special kind of polyhedral complexes: the smooth polyhedral complexes. We prove that the fundamental group of a smooth polyhedral complex of dimension n appears as the fundamental group of a log canonical singularity of dimension n+1. Given a 3-manifold M smoothly embedded in ℝ4, we show the existence of a smooth polyhedral complex of dimension 3 that is homotopic to M. To do so, we start from a complex homotopic to M and perform combinatorial modifications that mimic the resolution of singularities in algebraic geometry.