论文标题
计算定向标志复合物的同义类型
Computing Homotopy Types of Directed Flag Complexes
论文作者
论文摘要
组合和随机定义的简单络合物通常具有楔形楔形的同质类型。 Kahle的突出猜想将其量化为随机标志复合物的情况。我们探索此类属性是否可能扩展到由自然产生的图。我们考虑了秀丽隐杆线虫线虫的大脑网络（由Varshney＆Al。重建），这是生物学中重要的模型生物。使用基于代数拓扑基本方法的迭代计算程序，即同源性，简单崩溃和孔侧操作，我们表明其定向的旗配合物是同等的，等同于球形的楔形，这是第一次完全确定与大脑网络相对应的同质型号类型。 我们还考虑了相应的旗帜图，并表明扭转可以在其局部方向性过滤的同源性中找到。作为一个玩具的例子，麦凯系列的锦标赛的定向国旗综合体被归类为同型。除球体以外的摩尔空间中发生在此分类中。作为一种工具，我们证明，通过考虑其细胞结构，任何比赛的定向国旗综合体的基本组都是免费的。
Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such properties might extend to graphs arising from nature. We consider the brain network (as reconstructed by Varshney & al.) of the Caenorhabditis elegans nematode, an important model organism in biology. Using an iterative computational procedure based on elementary methods of algebraic topology, namely homology, simplicial collapses and coning operations, we show that its directed flag complex is homotopy equivalent to a wedge of spheres, completely determining, for the first time, the homotopy type of a flag complex corresponding to a brain network. We also consider the corresponding flag tournaplex and show that torsion can be found in the homology of its local directionality filtration. As a toy example, directed flag complexes of tournaments from McKay's collection are classified up to homotopy. Moore spaces other than spheres occur in this classification. As a tool, we prove that the fundamental group of the directed flag complex of any tournament is free by considering its cell structure.
