Fundamental groups of small simplicial complexes

TL;DR

This paper classifies the fundamental groups of small simplicial complexes with up to 8 vertices, providing a comprehensive list and examples, which aids in understanding their topological properties and applications.

Contribution

It offers the first complete classification of fundamental groups for complexes with up to 8 vertices and presents numerous examples for complexes with 9 vertices.

Findings

Complete list of fundamental groups for complexes with ≤8 vertices

Examples of fundamental groups for complexes with 9 vertices

Applications to topological conjectures and invariants

Abstract

The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a complete list of groups that arise as fundamental groups of simplicial complexes with at most $8$ vertices. In addition we give many examples of fundamental groups of complexes with $9$ vertices although the complete classification seems to be beyond reach at the moment. Our results lead to many applications, including progress on the Bj\"orner-Lutz conjecture regarding vertex-minimal triangulations of the Poincar\'e homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups.