通过面板结构看待多面体乘积的统一视角
A unifying view toward polyhedral products through panel structures
AI总结：
本文引入带面空间上的多面体乘积概念，为多面体乘积与广义矩角复形提供统一视角，通过计算其稳定分解、上同调环及等变上同调环，推广了面环概念，既重诠释了已知结果也推导了新结论。
摘要：
拓扑空间上的面板结构就是一族局部有限的闭子空间。一个配备面板结构的空间被称为带面空间。本文中，我们引入了带面空间上的多面体乘积概念。这一概念为不同场景下的多面体乘积构造与广义矩角复形（generalized moment-angle complexes）提供了统一的研究视角。我们计算了这些空间的稳定分解，并以此研究它们的上同调环结构。此外，我们能够计算带面空间上的矩角复形关于典范环面作用的等变上同调环。该计算引出了带面空间的拓扑面环概念，这一概念推广了单纯复形的经典面环（Stanley-Reisner 环）概念。我们将看到，多面体乘积与矩角复形研究中的许多已知结果都可以通过我们关于带面空间上多面体乘积的一般性定理得到重新诠释。此外，在某些场景下我们还能通过该方法推导出一些新的结果。
英文摘要：
A panel structure on a topological space is just a locally finite family of closed subspaces. A space together with a panel structure is called a space with faces. In this paper, we introduce a notion of polyhedral product over a space with faces. This notion provides a unifying viewpoint on the constructions of polyhedral products and generalized moment-angle complexes in various settings. We compute the stable decomposition of these spaces and use it to study their cohomology ring structures. Moreover, we can compute the equivariant cohomology ring of the moment-angle complex over a space with faces with respect to the canonical torus action. The calculation leads to the notion of topological face ring of a space with faces, which generalizes the classical notion of face ring (Stanley-Reisner ring) of a simplicial complex. We will see that many known results in the study of polyhedral products and moment-angle complexes can be reinterpreted from our general theorems on the polyhedral product over a space with faces. Moreover, we can derive some new results via our approach in some settings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
