Falk Invariants of Signed Graphic Arrangements
Falk Invariants of Signed Graphic Arrangements
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签名图形排列的 Falk 不变量
DOI：
10.1007/s00373-018-1958-9
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
G. Jiang
中科院分区：
文献类型：
--
作者：
Weili Guo;Qiumin Guo;G. Jiang
复向量空间中超平面排列的补的基本群是一个重要的拓扑不变量。第三级的连续的连续性在较低的中心系列的基本组被称为福尔克不变的安排，因为福尔克给了第一个组合公式，并要求给予组合的解释。本文证明了无圈带号图G的排列的Falk不变量是的和的两倍，其中是G中与有1个顶点的团交换等价的子图的个数，是有3个顶点且每两个顶点由不同符号的双边连接的子图的个数。这个公式修正了Schenck和Suciu给出的公式，并且部分地回答了Falk在带符号图排列情况下的问题。
The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first combinatorial formula and asked to give a combinatorial interpretation. In this article we prove that the Falk invariant of an arrangement associated with signed graphGwithout loops is double of the sum of,andwhereis the number of subgraphs ofGthat are switching equivalent to the cliques withlvertices,is that of subgraphs which have 3 vertices and each two vertices are connected by double edges with different signs. This formula modifies the one given by Schenck and Suciu, and answers partially Falk’s question in the case of signed graphic arrangements.