Subdivisions of lower Eulerian posets
Alan Stapledon
TL;DR
This work develops a rigorous bridge between strong formal subdivisions of lower Eulerian posets and canonical triples (Γ, ρ_Γ, q) via the non-Hausdorff mapping cylinder, establishing a categorical equivalence between Subdiv and JoinAdm^∘. It extends the theory to CW-posets, showing that the CYL–MAP correspondence respects CW-regularity and yields a CW-categorical equivalence, while providing concrete polytope and fan examples. A key application is a subdivision formula for the cd-index, linking local and global invariants and yielding nonnegativity results under Gorenstein* hypotheses. The framework opens avenues for Kazhdan–Lusztig–Stanley theory via a unifying subdivision perspective and suggests broad generalizations to locally Eulerian posets and beyond, with several open questions about obstructions, dualities, and higher categorical structures.
Abstract
There is a natural notion of a subdivision of a lower Eulerian poset called a strong formal subdivision, which abstracts the notion of a polyhedral subdivision of a polytope, or a proper, surjective morphism of fans. We show that there is a canonical bijection between strong formal subdivisions and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The bijection uses the non-Hausdorff mapping cylinder construction introduced by Barmak and Minian. A corresponding bijection for $CW$-posets is given, as well as an application to computing the $cd$-index of an Eulerian poset. A companion paper explores applications to Kazhdan-Lusztig-Stanley theory.