Problem
Research questions and friction points this paper is trying to address.
abstract simplicial complexes
stellar subdivisions
formalization
combinatorial topology
proof assistant
Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean
🤖 AI Summary
This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.
📝 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 $σ_s K$, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.
Innovation
Methods, ideas, or system contributions that make the work stand out.
abstract simplicial complexes
stellar subdivisions
formalization in Lean
combinatorial topology
links and joins
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