Fundamental groups of low-dimensional lc singularities

Abstract

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 R 4 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 R 4 , 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. Contents 1. Introduction 1 2. Preliminaries 6 3. Fundamental groups in dimension 2 8 4. Smooth polyhedral complexes 23 5. Threefold log canonical singularities 31 6. Fourfold log canonical singularities 36 7. Examples and questions 42 Appendix A. Fundamental groups of lc surface singularities 44 References 46