  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.
