标题： 通过超凸嵌入，$(\mathbb{R}^n, \ell_1)$中稠密子集的 Vietoris-Rips 复形的收缩性 显示英文标题

标题： Contractibility of Vietoris-Rips Complexes of dense subsets in $(\mathbb{R}^n, \ell_1)$ via hyperconvex embeddings

摘要： 我们考虑在足够大的尺度下，$(\mathbb{R}^n,\ell_1)$密子集的 Vietoris-Rips 复形的可缩性。 这是由 Matthew Zaremsky 提出的一个问题所激发的，即对于每个$n$自然数，是否存在一个$r_n>0$，使得$(\mathbb{Z}^n,\ell_1)$在尺度$r$的 Vietoris-Rips 复形对于所有$r\geq r_n$都是可缩的。 我们通过与度量空间$X$嵌入到超凸度量空间的邻域相关的结果以及其与 Vietoris-Rips 复形$X$的联系来解决这个问题。以这种方式，我们对情况$n=2$和$3$给出了肯定的回答。 显示英文摘要

摘要： We consider the contractibility of Vietoris-Rips complexes of dense subsets of $(\mathbb{R}^n,\ell_1)$ with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each $n$ natural there is a $r_n>0$ so that the Vietoris-Rips complex of $(\mathbb{Z}^n,\ell_1)$ at scale $r$ is contractible for all $r\geq r_n$. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space $X$ and its connection to the Vietoris-Rips complex of $X$. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$.