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!
|
| Summary: | 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. |
|---|---|
| ISSN: | 2050-5094 |