Rigidity of symmetric simplicial complexes and the lower bound theorem
We show that if $\Gamma $ is a point group of $\mathbb {R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal {S}$ is a k-pseudomanifold which has a free automorphism of order two, then either $\mathcal {S}$ has a $\Gamma $ -symmetric infinitesimally rigid...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001506/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832594076728295424 |
|---|---|
| author | James Cruickshank Bill Jackson Shin-ichi Tanigawa |
| author_facet | James Cruickshank Bill Jackson Shin-ichi Tanigawa |
| author_sort | James Cruickshank |
| collection | DOAJ |
| description | We show that if
$\Gamma $
is a point group of
$\mathbb {R}^{k+1}$
of order two for some
$k\geq 2$
and
$\mathcal {S}$
is a k-pseudomanifold which has a free automorphism of order two, then either
$\mathcal {S}$
has a
$\Gamma $
-symmetric infinitesimally rigid realisation in
${\mathbb R}^{k+1}$
or
$k=2$
and
$\Gamma $
is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when
$\Gamma $
is a point-inversion group. Our result implies that Stanley’s lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial automorphism of order 2, thus verifying (the inequality part of) another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes – namely, the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes. |
| format | Article |
| id | doaj-art-6a7c257829714607866a6c68779ea96d |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-6a7c257829714607866a6c68779ea96d2025-01-20T06:08:11ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.150Rigidity of symmetric simplicial complexes and the lower bound theoremJames Cruickshank0Bill Jackson1Shin-ichi Tanigawa2School of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, H91 TK33, Ireland; E-mail:School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United KingdomDepartment of Mathematical Informatics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8654, Japan; E-mail:We show that if $\Gamma $ is a point group of $\mathbb {R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal {S}$ is a k-pseudomanifold which has a free automorphism of order two, then either $\mathcal {S}$ has a $\Gamma $ -symmetric infinitesimally rigid realisation in ${\mathbb R}^{k+1}$ or $k=2$ and $\Gamma $ is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when $\Gamma $ is a point-inversion group. Our result implies that Stanley’s lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial automorphism of order 2, thus verifying (the inequality part of) another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes – namely, the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.https://www.cambridge.org/core/product/identifier/S2050509424001506/type/journal_article52C2505E4505C10 |
| spellingShingle | James Cruickshank Bill Jackson Shin-ichi Tanigawa Rigidity of symmetric simplicial complexes and the lower bound theorem Forum of Mathematics, Sigma 52C25 05E45 05C10 |
| title | Rigidity of symmetric simplicial complexes and the lower bound theorem |
| title_full | Rigidity of symmetric simplicial complexes and the lower bound theorem |
| title_fullStr | Rigidity of symmetric simplicial complexes and the lower bound theorem |
| title_full_unstemmed | Rigidity of symmetric simplicial complexes and the lower bound theorem |
| title_short | Rigidity of symmetric simplicial complexes and the lower bound theorem |
| title_sort | rigidity of symmetric simplicial complexes and the lower bound theorem |
| topic | 52C25 05E45 05C10 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001506/type/journal_article |
| work_keys_str_mv | AT jamescruickshank rigidityofsymmetricsimplicialcomplexesandthelowerboundtheorem AT billjackson rigidityofsymmetricsimplicialcomplexesandthelowerboundtheorem AT shinichitanigawa rigidityofsymmetricsimplicialcomplexesandthelowerboundtheorem |