$B$-rigidity of ideal almost Pogorelov polytopes
$B$-rigidity of ideal almost Pogorelov polytopes
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$B$-理想的几乎 Pogorelov 多面体的刚性
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
N. Erokhovets
中科院分区：
文献类型：
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作者：
N. Erokhovets
复曲面拓扑赋予每个$n$维组合简单凸多面体$P$与$m$方面的$（m+n）$维矩角流形$\mathcal{Z}_P$与一个紧凑的环面$T^m$的作用，使$\mathcal{Z}_P/T^m$是一个凸多面体的组合类型$P$。一个单$n$-多胞形称为$B$-刚性的，如果对于一个单$n$-多胞形$Q$，分次环$H^*（\mathcal {Z}_P，\mathbb Z）= H^*（\mathcal{Z}_Q，\mathbb Z）$的任何同构蕴涵$P$和$Q$是组合等价的.理想几乎Pogorelov多面体是一个组合的3 $-多面体，它是在Lobachevsky（双曲）空间$\mathbb L^3$中切掉一个理想直角多面体的所有理想顶点而得到的。这些多面体正是从任何不一定简单的凸3 $-多面体通过切除所有顶点然后切除所有“旧”边而得到的多面体。对偶多胞形的边界是旧多胞形（也是它的对偶多胞形）边界的重心细分。证明了理想几乎Pogorelov多面体是$B$-刚性的.这产生了理想几乎Pogorelov流形上的三个上同调刚性流形族：矩角流形，规范$6$维拟心流形和规范$3$维小覆盖，它们是“从线性模型拉回”。
Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold $\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that $\mathcal{Z}_P/T^m$ is a convex polytope of combinatorial type $P$. A simple $n$-polytope is called $B$-rigid, if any isomorphism of graded rings $H^*(\mathcal{Z}_P,\mathbb Z)= H^*(\mathcal{Z}_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that $P$ and $Q$ are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial $3$-polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space $\mathbb L^3$. These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex $3$-polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is $B$-rigid. This produces three cohomologically rigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical $6$-dimensional quasitoric manifolds and canonical $3$-dimensional small covers, which are "pullbacks from the linear model".
简单多面体的环面上同调刚度
DOI：
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发表时间：
2010
期刊：
J.London Math.Soc
影响因子：
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作者：
S.Choi;T.Panov;D.Y.Suh
通讯作者：
D.Y.Suh