Topological complexity of a map
Topological complexity of a map
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地图的拓扑复杂度
DOI：
10.4310/hha.2019.v21.n2.a7
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发表时间：
2018
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影响因子：
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通讯作者：
Petar Pavešić
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文献类型：
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作者：
Petar Pavešić
我们研究了某些受自主机器人操纵应用启发的拓扑问题。考虑连续映射 $f\colon X\to Y$，其中 $f$ 可以是从机器人手臂或类似机构的配置空间 $X$ 到工作空间 $Y$ 的运动学映射。然后我们可以将 $f$ 与一个数字 $\mathrm{TC}(f)$ 关联起来，粗略地说，这是为设备构建完整的操作算法所需的最小连续规则数。示例表明 $\mathrm{TC}(f)$ 对 $f$ 的小扰动非常敏感，并且其值在很大程度上取决于 $f$ 的奇点。这一事实使计算变得相当复杂，因此我们在这里关注 $\mathrm{TC}(f)$ 的估计，这些估计可以用空间 $X$ 和 $Y$ 的同伦不变量来表示，或者如果 $f$ 满足一些额外的假设（例如纤维化），则该估计是有效的。
一些主要结果是$\mathrm{TC}(f)$的一般上限的推导、$\mathrm{TC}(f)$相对于域和余域变形的不变性、证明$\mathrm{TC}(f)$是FHE不变量以及$\mathrm{TC}(f)$的上同调下界的描述。此外，如果 $f$ 是纤维化，我们根据 Lusternik-Schnirelmann 类别以及 $X$ 和 $Y$ 的拓扑复杂度得出对 $\mathrm{TC}(f)$ 更精确的估计。我们还获得了覆盖投影这一重要特例的一些结果。
We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f\colon X\to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$ of a robot arm or a similar mechanism. Then one can associate to $f$ a number $\mathrm{TC}(f)$, which is, roughly speaking, the minimal number of continuous rules that are necessary to construct a complete manipulation algorithm for the device. Examples show that $\mathrm{TC}(f)$ is very sensitive to small perturbations of $f$ and that its value depends heavily on the singularities of $f$. This fact considerably complicates the computations, so we focus here on estimates of $\mathrm{TC}(f)$ that can be expressed in terms of homotopy invariants of spaces $X$ and $Y$, or that are valid if $f$ satisfy some additional assumptions like, for example, being a fibration.
Some of the main results are the derivation of a general upper bound for $\mathrm{TC}(f)$, invariance of $\mathrm{TC}(f)$ with respect to deformations of the domain and codomain, proof that $\mathrm{TC}(f)$ is a FHE-invariant, and the description of a cohomological lower bound for $\mathrm{TC}(f)$. Furthermore, if $f$ is a fibration we derive more precise estimates for $\mathrm{TC}(f)$ in terms of the Lusternik-Schnirelmann category and the topological complexity of $X$ and $Y$. We also obtain some results for the important special case of covering projections.