Subdividing the cd-index of Eulerian Posets

TL;DR

This thesis explores the subdivision properties of the cd-index in Eulerian posets, introduces a local cd-index concept, and extends subdivision decomposition results to broader classes of complexes and polytopes.

Contribution

It provides a combinatorial proof of the cd-index subdivision decomposition, defines a new local cd-index, and connects these concepts to h-vector bounds and poset structures.

Findings

Extended subdivision decomposition to wider classes of complexes.

Introduced the local cd-index analogous to local h-vector.

Bound the cd-index of certain polytopes using the local cd-index.

Abstract

This thesis aims to give the reader an introduction and overview of the cd-index of a poset, as well as establish some new results. We give a combinatorial proof of Ehrenborg and Karu's cd-index subdivision decomposition for Gorenstein* complexes and extend it to a wider class of subdivisions. In doing so, we define a local cd-index that behaves analogously to the well studied local h-vector. We examine known cd-index and h-vector bounds, and then use the local cd-index to bound a particular class of polytopes with the cd-index of a stacked polytope. We conclude by investigating the h-vector and local h- vector of posets in full generality, and use an algebra morphism developed by Bayer and Ehrenborg to demonstrate the structural connection between the cd-index subdivision decomposition and the local h-vector subdivision decomposition. iii