Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean

Garett Cunningham; Daniel Zach; Stefan Friedl

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arXiv · 2607.10216

Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean

Garett Cunningham·Daniel Zach·Stefan Friedl

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Abstract

The theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex $K$ and in a stellar subdivision $\sigma_s K$, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.

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Garett Cunningham, Daniel Zach, Stefan Friedl. 2026-07-11. Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean. https://arxiv.org/abs/2607.10216

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