Discrete Morse theory and graph braid groups
Discrete Morse theory and graph braid groups
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离散莫尔斯理论和图辫群
DOI：
10.2140/agt.2005.5.1075
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发表时间：
2004
影响因子：
0.7
通讯作者：
Lucas Sabalka
中科院分区：
文献类型：
--
作者：
Daniel S. Farley;Lucas Sabalka
如果是任意有限图，则由n个点组成的无标号配置空间，记为U C n，是的n元子集空间。N股的辫子群是U C n的基本群。我们将Morse理论的一个离散版本应用于这些U C n，对于任意n和任意n，给出了每种情况下的临界胞格的清晰描述。因此，我们可以为任意数量的线束计算任意树的辫子群的表示。我们还给出了Ghrist的一个定理的简单证明：空间U C n强变形收缩到至多k维的CW复形上，其中k是至少3次的顶点数(因此k与n无关)。AMS分类20F65、20F36；57M15、57Q05、55R80
If is any finite graph, then the unlabelled configuration space of n points on , denoted U C n , is the space of n-element subsets of . The braid group of on n strands is the fundamental group of U C n . We apply a discrete version of Morse theory to these U C n , for any n and any , and provide a clear description of the critical cells i n every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space U C n strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in of degree at least 3 (and k is thus independent of n). AMS Classification 20F65, 20F36; 57M15, 57Q05, 55R80