Face Numbers of Shellable CW Balls and Spheres

TL;DR

This paper extends lower bounds on face numbers of shellable CW spheres and balls, generalizing previous polytope results and establishing conditions for equality and minimal face counts across dimensions.

Contribution

It proves new inequalities for face numbers of shellable CW spheres and balls, including a generalization of Baryany's question, and characterizes cases of equality and minimal face counts.

Findings

Established lower bounds for face numbers of shellable CW spheres and balls.

Characterized equality cases when the bounds are tight.

Proved minimal face number conditions for CL-shellable CW spheres.

Abstract

Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $\rho(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering B\'ar\'any's question from 1998, proved that for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[ f_k(\mathscr{X}) \geq \rho(d+1,k)f_d(\mathscr{X}). \] We prove a generalization: if $\mathscr{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[ f_k(\mathscr{X}) \geq \rho(d+1,k)f_d(\mathscr{X}) + \frac{1}{2}f_k(\partial \mathscr{X}), \] with equality precisely when $k=d$ or when $k=d-1$ and $\mathscr{X}$ is simplicial. We further prove that if $\mathscr{S}$ is a strongly regular CW sphere of dimension $d$, and the face poset of $\mathscr{S}$ is both CL-shellable and dual CL-shellable, then $f_k(\mathscr{S}) \geq \min\{f_0(\mathscr{S}),f_d(\mathscr{S})\}$ for all $0 \leq k \leq d$.