标题： 球面的Čech复形的同伦连通性 显示英文标题

标题： Homotopy connectivity of Čech complexes of spheres

摘要： 设$S^n$为带有测地度量且直径为$\pi$的$n$-球面。 $S^n$在尺度$r$下的固有Čech复形是所有半径为$r$的开球的神经，这些开球位于$S^n$中。 In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. 我们的界表明，对于$n\ge 1$，在尺度$r$下，$S^n$的Čech复形的同伦类型随着$r$在$(0,\pi)$中变化而无限多次改变；我们猜想仅发生可数次这样的变化。 此外，我们根据它们的包装情况，对$S^n$的有限子集的Čech复形的同调维数给出了下界。 显示英文摘要

摘要： Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic \v{C}ech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of \v{C}ech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \v{C}ech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the \v{C}ech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,\pi)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of \v{C}ech complexes of finite subsets of $S^n$ in terms of their packings.