Stellar theory for flag complexes

TL;DR

This paper characterizes when two flag complexes are piecewise linearly homeomorphic using edge subdivisions and explores conjectures on the combinatorial structure of flag spheres based on face numbers.

Contribution

It refines Alexander's result for flag complexes and introduces new conjectures on flag sphere structures, along with an algorithm to test these conjectures.

Findings

Two flag complexes are PL homeomorphic iff connected by edge subdivisions.

Proposes new conjectures on face number constraints for flag spheres.

Algorithm can exhaustively test conjectures on flag PL spheres.

Abstract

Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge subdivision or its inverse. For flag spheres we pose new conjectures on their combinatorial structure forced by their face numbers, analogous to the extremal examples in the upper and lower bound theorems for simplicial spheres. Furthermore, we show that our algorithm to test the conjectures searches through the entire space of flag PL spheres of any given dimension.