论文标题
随机复合物，有自由互动
Random complexes with free involution
论文作者
论文摘要
我们引入了一个新的模型，用于随机简单络合物，该模型具有高概率，生成了具有简单连接双盖的复合物。因此，我们开发了一个使用基本组$ \ mathbb {z}/2 \ mathbb {z} $的随机简单复合物的模型。我们建立了有关这些复合物的典型渐近拓扑结构的结果。结果，我们给出了尺寸$ d $的界限，以使$ \ mathbb {z}/2 \ mathbb {z} $ - 从双层盖到$ \ mathbb {r}^d $具有较高可能性的Zeros，从而建立了随机的borsuk-ulam theorem。我们将其应用于erdős--rényi随机图中的成对的非粘合物簇的结构结果。
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural result for pairs of non-adjacent cliques in Erdős--Rényi random graphs.
