Fundamental groups of small simplicial complexes

TL;DR

The paper tackles the problem of which abstract groups appear as fundamental groups of simplicial complexes on a given number of vertices, achieving a complete classification for complexes with at most eight vertices. It develops a structural decomposition and a scalable algorithm to enumerate all realizable groups up to free factors, and it demonstrates new 9-vertex examples and tight implications for vertex-minimal triangulations of the Poincaré sphere. The results yield precise vertex bounds for triangulations of homology spheres, sharpen recognition criteria for PL triangulations, and extend the Karoubi–Weibel invariant computations for a broad class of groups. The methods connect combinatorial topology with group-theoretic invariants, providing a practical toolkit for examining minimal triangulations and aspherical spaces.

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 Björner-Lutz conjecture regarding vertex-minimal triangulations of the Poincaré homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups.

Figures (4)

• Figure 1: The complex $Y=(X_1\times_{\mathrm{CW}}X_2)^{(2)}$ can be pictured as a triangular prism in $\mathbb{R}^3$, subdivided into five rooms. Two of these rooms are shaped like tetrahedra, while the other three are shaped like triangular prisms.
• Figure 2: The minimal triangulation of a real projective plane $\mathbb{R} P^2$.
• Figure 3: The minimal triangulation of a torus --- Császár torus.
• Figure 4: Simplicial complex on $8$ vertices with the fundamental group isomorphic to the braid group on $3$ strands.

Theorems & Definitions (40)

• Theorem 2.1: Qian
• Remark 2.2
• Theorem 3.1: C. T. C. Wall Wall, Proposition 3.3
• Proposition 3.2
• proof
• Remark 3.3
• Proposition 3.4
• Proposition 3.5
• proof
• Theorem 3.6