Database of fundamental groups of small simplicial complexes

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 dataset, we include triangulations on a small number of vertices (mainly 8 or 9) with given fundamental groups.
More precisely, it consists of three text files: KW(G)=8.txt, KW(G)=9.txt and Free_Abelian_Complexes.txt, that contain:

KW(G)=8.txt: a complete list of groups (up to free product with a free group) that arise as fundamental groups of simplicial complexes on 8 vertices but not fewer, together with example triangulations. This provides a classification, up to free factor, of groups G with Karoubi-Weibel invariant KW(G) equal to 8,
KW(G)=9.txt: many new examples of groups G that appear as fundamental groups of complexes on 9 vertices but not fewer, so that KW(G)=9,
Free_Abelian_Complexes.txt: reasonably small triangulations for free abelian groups of rank 3, 4, 5 and 6.

Our results lead to many applications, including the computation of the Karoubi–Weibel invariant for many groups, progress on the Björner–Lutz conjecture regarding vertex-minimal triangulations of the Poincaré homology sphere and improved recognition criteria for PL-triangulations of manifolds.
The dataset was produced using SageMath and a C++ program. Detailed information, such as the description of the algorithm, computations and theoretical background, can be found in:
D. Govc, W. Marzantowicz, Ł. P. Michalak and P. Pavešić, Fundamental groups of small simplicial complexes, preprint 2025.