Using the ordered analogue of Farley–Sabalka’s discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber’s equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.

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Farley-Sabalka 的 Morse 理论模型和树上有序配置空间的更高拓扑复杂性

在图的配置空间上使用 Farley-Sabalka 离散梯度场的有序模拟，我们解开了树上纯编织群的生成器的水平行为。这使我们能够概括法伯对两个粒子上的树上配置空间的同伦类型的等变描述。结果适用于计算任意数量粒子上树上有序​​配置空间的所有更高拓扑复杂性。