论文标题
某些多面体产物的共同体的模型
Models for the Cohomology of Certain Polyhedral Products
论文作者
论文摘要
对于带有单位的交换环$ \ Mathbf k $，我们描述和研究各种差异分级$ \ mathbf k $ - 模块和$ \ Mathbf K $ -Algebras，它们是多面体产品共同体$（\ useverline {cx} {cx} {cx}，\ usevenlline x）^k $的模型。在此过程中，我们证明了真实的Moment-angle-never-necemology $ h^*（（d^1，s^0）^k; \ mathbb z）$ the Moment-Moment-angle-necle-nemle angle-neve confffect是一个模块，它不是来自几何设置的模块。我们还揭示了$ h^*（（d^1，s^0）^k; \ mathbb z）$和$ h^**（（d^n，s^n，s^{n-1}）^k; \ mathbb z）$ for $ n \ geq 2 $具有相同的来源。 作为应用程序，这项工作为研究$（\ usepline {cx}，\下划线x）^k $的基于循环空间的阶段奠定了阶段$ \ MATHCAL C^*（（\下划线{Cx}，\下划线X）^K; \ Mathbb Z）$。
For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K; \mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does not come from a geometric setting. We also reveal that the apriori different cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n, S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin. As an application, this work sets the stage for studying the based loop space of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X; \mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$.
