论文标题
$ \ mathbb {s}^1 $的曲率集的同源组
Homology groups of the curvature sets of $\mathbb{S}^1$
论文作者
论文摘要
对于$ n \ geq 2 $，$ n $ th曲率集$ x $是由所有$ n $ by-by- $ n $ n $ danks矩阵组成的集合，$ n $ n $ suption $ n $ points均从$ x $采样。曲率集可以被视为配置空间的几何类似物。在本文中，我们对配备了测量指标的单元圆$ \ mathbb {s}^1 $进行了几何和拓扑研究。通过归纳参数，我们计算$ \ mathbb {s}^1 $的所有曲率集的同源组。我们还构建了一个抽象的简单复合物，称为$ n $ th State Complex，其几何实现是$ \ Mathbb {S}^1 $的$ n $ Th曲率集的同型。
For $n \geq 2$, the $n$-th curvature set of a metric space $X$ is the set consisting of all $n$-by-$n$ distance matrices of $n$ points sampled from $X$. Curvature sets can be regarded as a geometric analogue of configuration spaces. In this paper we carry out a geometric and topological study of the curvature sets of the unit circle $\mathbb{S}^1$ equipped with the geodesic metric. Via an inductive argument we compute the homology groups of all curvature sets of $\mathbb{S}^1$. We also construct an abstract simplicial complex, called the $n$-th State Complex, whose geometric realization is homeomorphic to the $n$-th Curvature Set of $\mathbb{S}^1$.
