Fundamental groups of small simplicial complexes

2025/11/04 by Govc, Dejan, Marzantowicz, Wacław, Michalak, Łukasz Patryk +1
#20F65 (Primary) 55M30 #55U10 #57-04 #57Q15 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.02586

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.

Citations

• On the number of inequivalent monotone Boolean functions of 9 variables
• A computation of the ninth Dedekind Number
• The Karoubi-Weibel complexity for groups
• Hadamard Matrix Torsion
• Vertex numbers of simplicial complexes with free abelian fundamental group
• A survey of the foundations of four-manifold theory in the topological category
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• Triangulations with few vertices of manifolds with non-free fundamental group
• The covering type of closed surfaces and minimal triangulations
• Estimates of covering type and the number of vertices of minimal triangulations
• Small simplicial complexes with prescribed torsion in homology
• On the covering type of a space
• Systolic invariants of groups and 2-complexes via Grushko decomposition

Related

• Vertex numbers of simplicial complexes with free abelian fundamental group
• Fundamental Groups of Simplicial Complexes
• Computing Invariants of Simplicial Manifolds
• Vertex-Minimal Triangulation of Complexes with Homology
• Small simplicial complexes with prescribed torsion in homology