Polyhedral products over finite posets
Polyhedral products over finite posets
复制标题
DOI：
10.1215/21562261-2022-0020
复制
发表时间：
2019-03
影响因子：
0.6
通讯作者：
D. Kishimoto;R. Levi
中科院分区：
文献类型：
--
作者：
D. Kishimoto;R. Levi
巴赫里、本德斯基、科恩和吉特勒将多面体乘积定义为以抽象单纯复形的单纯形为指标的某些乘积空间的并。本文给出了任意点偏序集上多面体乘积的一个非常一般的同伦理论构造。我们证明了在偏序集$\CALP$的某些限制下，包括所有已知的情形，所得到的空间的上同调可以计算为构造块的上同调在$\CALP$上的逆极限。这激发了一种类似的代数结构的定义--多面体张量积。我们证明了对于一个大的偏序集族，多面体乘积的上同调由多面体张量积给出。然后我们将注意力限制在多面体偏序集上，这是一族偏序集族，包括单纯复形的面偏序集和单纯偏序集，以及许多其他偏序集。我们定义了多面体偏序集的Stanley-Reisner环，并证明了，与经典情形一样，这些环是该偏序集上某些多面体乘积的上同调。对于任意点偏序集$\CALP$，我们构造了一个单纯偏序集$S(\CALP)$，并证明了如果$\CALP$是多面体偏序集，则$\CALP$上的多面体乘积与相应的$S(\CALP)$上的多面体乘积同伦.
Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $\calp$, that include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $\calp$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction - the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that include face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley-Reisner ring of a polyhedral poset and show that, like in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $\calp$ we construct a simplicial poset $s(\calp)$, and show that if $\calp$ is a polyhedral poset then polyhedral products over $\calp$ coincide up to homotopy with the corresponding polyhedral products over $s(\calp)$.