论文标题
地图的较高拓扑复杂性
Higher topological complexity of a map
论文作者
论文摘要
空间$ x $，$ \ text {tc} _r（x）$，$ r = 2,3，\ ldots $以及地图$ f $，$ f $，$ \ text {tc}（f）$的较高拓扑复杂性，以及远处的扩展性，分别由rudyak andpavešić引入了Aptermension。在本文中，我们介绍了地图〜$ f $，$ \ text {tc} _ {r，s}（f）$的较高拓扑复杂性的概念，以$ 1 \ leq s \ leq s \ leq r \ geq2 $，同时扩展了Rudyyak's andPavešić的概念。当向前运动学任务的$ S $规定阶段中扮演角色时，我们统一的概念与机器人相关的$ r $ - 多义务运动计划问题相关。我们在地图的产品和构图下研究$ \ text {tc} _ {tc} _ {tc} _ {tc} _ {tc} $的行为，以及$ \ text {tc} _ {tc} _ {r，s} $在$ r $和$ s $上的依赖性。我们在与$ x $，$ y $和$ f $相关的分类不变性方面绘制$ \ text {tc} _ {r，s} _ {r，s} $ $ text {特别是，在一个$ \ text {tc} _ {r，s} $的值中，在一个非平凡的双重覆盖物以及其复杂的对应物的情况下。
The higher topological complexity of a space $X$, $\text{TC}_r(X)$, $r=2,3,\ldots$, and the topological complexity of a map $f$, $\text{TC}(f)$, have been introduced by Rudyak and Pavešić, respectively, as natural extensions of Farber's topological complexity of a space. In this paper we introduce a notion of higher topological complexity of a map~$f$, $\text{TC}_{r,s}(f)$, for $1\leq s\leq r\geq2$, which simultaneously extends Rudyak's and Pavešić's notions. Our unified concept is relevant in the $r$-multitasking motion planning problem associated to a robot devise when the forward kinematics map plays a role in $s$ prescribed stages of the motion task. We study the homotopy invariance and the behavior of $\text{TC}_{r,s}$ under products and compositions of maps, as well as the dependence of $\text{TC}_{r,s}$ on $r$ and $s$. We draw general estimates for $\text{TC}_{r,s}(f\colon X\to Y)$ in terms of categorical invariants associated to $X$, $Y$ and $f$. In particular, we describe within one the value of $\text{TC}_{r,s}$ in the case of the non-trivial double covering over real projective spaces, as well as for their complex counterparts.
