论文标题
顺序参数化的拓扑复杂性和相关不变性
Sequential parametrized topological complexity and related invariants
论文作者
论文摘要
参数化运动计划算法\ cite {cfw}具有高度的普遍性和灵活性；它们在各种外部条件下产生机器人系统的运动。后者被视为参数，构成算法输入的一部分。顺序参数化拓扑复杂性$ {\ sf tc} _r [p：e \ to b] $的概念是对这种算法的复杂性的度量。它在\ cite {cfw，cfw2}中以$ r = 2 $和\ cite {fp}的价格进行了研究，以$ r \ ge 2 $。在本文中，我们分析了复杂性的依赖性$ {\ sf tc} _r [p：e \ to b] $在带有结构组$ g $的初始捆绑包上，并在其光纤$ x $上被视为$ g $空加。我们的主要结果估计$ {\ sf tc} _r [p：e \ to b] $在捆绑包的某些不变和光纤上的动作方面。此外，我们还根据底座和光纤获得估计。最后，我们开发了一个截面类别的演算，其中包含一个新的不变$ {\ sf secat} _f [p：e \ to b] $，该$在纤维塔截面类别的研究中起着重要作用。
Parametrized motion planning algorithms \cite{CFW} have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions. The latter are viewed as parameters and constitute part of the input of the algorithm. The concept of sequential parametrized topological complexity ${\sf TC}_r[p:E\to B]$ is a measure of the complexity of such algorithms. It was studied in \cite{CFW, CFW2} for $r=2$ and in \cite{FP} for $r\ge 2$. In this paper we analyse the dependence of the complexity ${\sf TC}_r[p:E\to B]$ on an initial bundle with structure group $G$ and on its fibre $X$ viewed as a $G$-space. Our main results estimate ${\sf TC}_r[p:E\to B]$ in terms of certain invariants of the bundle and the action on the fibre. Moreover, we also obtain estimates depending on the base and the fibre. Finally, we develop a calculus of sectional categories featuring a new invariant ${\sf secat}_f[p:E\to B]$ which plays an important role in the study of sectional category of towers of fibrations.
