CW posets after the Poincare Conjecture

TL;DR

This paper discusses how the proof of the Poincare Conjecture allows for a broader set of criteria to identify CW posets, replacing shellability with homotopy Cohen-Macaulayness.

Contribution

It extends Bjorner's characterization of CW posets by replacing shellability with homotopy Cohen-Macaulayness, broadening the methods for proving a poset is a CW poset.

Findings

Homotopy Cohen-Macaulay property can replace shellability in CW poset characterization

The proof of the Poincare Conjecture broadens tools for identifying CW posets

Expanded criteria facilitate new applications in topological combinatorics

Abstract

Anders Bjorner characterized which finite graded partially ordered sets arise as the posets of closure relations on cells of a finite, regular CW complex. His characterization of these "CW posets" required each open interval $(\hat{0},u)$ to have order complex homeomorphic to a sphere of dimension $rk(u)-2$. Work of Danaraj and Klee showed that sufficient conditions were for the poset to be thin and shellable. The proof of the Poincare Conjecture enables the requirement of shellability to be replaced by the homotopy Cohen-Macaulay property. This expands the range of tools that may be used to prove a poset is a CW poset.