Face Numbers of Shellable CW Balls and Spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910602083958784 |
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| author | Hinman, Joshua |
| author_facet | Hinman, Joshua |
| contents | Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $ρ(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering Bárány's question from 1998, proved that for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[
f_k(\mathscr{X}) \geq ρ(d+1,k)f_d(\mathscr{X}). \] We prove a generalization: if $\mathscr{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[
f_k(\mathscr{X}) \geq ρ(d+1,k)f_d(\mathscr{X}) + \frac{1}{2}f_k(\partial \mathscr{X}), \] 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 $0 \leq k \leq d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Face Numbers of Shellable CW Balls and Spheres Hinman, Joshua Combinatorics 52B05 (Primary), 57Q05 (Secondary), 06A07 (Secondary) Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $ρ(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering Bárány's question from 1998, proved that for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[ f_k(\mathscr{X}) \geq ρ(d+1,k)f_d(\mathscr{X}). \] We prove a generalization: if $\mathscr{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, \[ f_k(\mathscr{X}) \geq ρ(d+1,k)f_d(\mathscr{X}) + \frac{1}{2}f_k(\partial \mathscr{X}), \] 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 $0 \leq k \leq d$. |
| title | Face Numbers of Shellable CW Balls and Spheres |
| topic | Combinatorics 52B05 (Primary), 57Q05 (Secondary), 06A07 (Secondary) |
| url | https://arxiv.org/abs/2409.08427 |