simpcomp -- A GAP toolbox for simplicial complexes

simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of manifolds.

💡 Research Summary

The paper presents “simpcomp”, a GAP package that brings comprehensive simplicial‑complex functionality into the GAP computer algebra system. While the existing GAP package “homology” focuses narrowly on simplicial homology, simpcomp treats a simplicial complex as a first‑class GAP object, encapsulating its facets, automorphism group, dimension, and a host of derived invariants. Implemented entirely in GAP’s scripting language, the package is fully transparent: users can inspect, modify, or extend any routine directly in GAP.

Simpcomp’s capabilities fall into three broad categories. First, it provides a rich set of constructors for standard complexes (simplices, cross‑polytopes, projective spaces, etc.) and for user‑defined complexes via facet lists, generator sets, or prescribed automorphism groups. The latter is a unique feature: given a group action and a small set of seed simplices, the whole complex can be generated automatically, which is especially useful when a triangulation is described in the literature only up to symmetry.

Second, the package implements combinatorial transformations. Bistellar (Pachner) moves, random sphere generation, and normal‑surface slicings are available, enabling heuristic checks of the combinatorial‑manifold property and the construction of minimal‑vertex triangulations through simulated‑annealing strategies. The authors highlight a heuristic algorithm (originally due to Lutz and Björner) that uses bistellar flips to test whether a complex is a combinatorial manifold, a task that would be cumbersome to code from scratch.

Third, simpcomp computes a wide array of invariants: the f‑vector, Euler characteristic, homology groups, fundamental group (via GAP’s group facilities), intersection‑form parity and signature, tightness, and perfect Morse‑function data. The paper demonstrates these features on the celebrated 16‑vertex triangulation of a K3 surface. By constructing the complex from its automorphism group and two seed simplices, the authors obtain the f‑vector