Fundamental groups of low-dimensional lc singularities

TL;DR

This paper investigates the fundamental groups of low-dimensional log canonical singularities, revealing their structure and realizing various groups as fundamental groups in specific dimensions, using novel polyhedral complexes.

Contribution

It introduces smooth polyhedral complexes and demonstrates their role in realizing fundamental groups of lc singularities across dimensions 2 to 4.

Findings

Fundamental groups of 2D lc singularities are finite extensions of solvable groups.

Every surface group appears as a fundamental group of a 3D lc singularity.

Every free group is realizable as the fundamental group of a 4D lc singularity.

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 $r\geq 2$ the free group $F_r$ 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 $\mathbb{R}^4$ 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 $\mathbb{R}^4$, 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.