论文标题
关于多面体产品的结构
On the Structure of Polyhedral Products
论文作者
论文摘要
在本文中，我们研究了多面体产品$ \ MATHCAL {z} _ {\ MATHCAL {k}}}（d^1，s^0）$由抽象的简单复杂$ {\ Mathcal {k}} $确定的结构。我们表明，HyperCube图在$ \ Mathcal {z} _ {\ Mathcal {k} _n}（d^1，s^0）$中有自然嵌入。这也提供了有关HyperCube图属的已知定理的新证明。我们描述了多面体产品函子的可逆自然变化。然后，我们研究了循环组$ \ mathbb {z} _n $在空间$ \ mathcal {z} _ {\ mathcal {k} _n}（d^1，s^0）$上的动作。此操作确定$ \ MATHBB {Z} [\ MATHBB {z} _n] $ - 同源性组的模块结构$ h _*（\ Mathcal {Z} _ {\ Mathcal {k}我们还研究了与同质轨道空间相关的leray-serre光谱序列$ e \ mathbb {z} _n \ times _ {\ mathbb {z} _n} \ mathcal {z} _ {z} _ {\ mathcal {\ mathcal {k} _n} _n} _n} _n}（d^1^1 s^0）$。
In this thesis, we study the structure of the polyhedral product $\mathcal{Z}_{\mathcal{K}}(D^1,S^0)$ determined by an abstract simplicial complex ${\mathcal{K}}$ and the pair $(D^1,S^0)$. We showed that there is natural embedding of the hypercube graph in $\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$ where ${\mathcal{K}}_n$ is the boundary of an $n$-gon. This also provides a new proof of a known theorem about genus of the hypercube graph. We give a description of the invertible natural transformations of the polyhedral product functor. Then, we study the action of the cyclic group $\mathbb{Z}_n$ on the space $\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$. This action determines a $\mathbb{Z}[\mathbb{Z}_n]$-module structure of the homology group $H_*(\mathcal{Z}_{\mathcal{K}_n}(D^1,S^0))$. We also study the Leray-Serre spectral sequence associated to the homotopy orbit space $E\mathbb{Z}_n\times_{\mathbb{Z}_n} \mathcal{Z}_{\mathcal{K}_n}(D^1,S^0)$.
