无限带中圆盘的位形空间
Configuration spaces of disks in an infinite strip
摘要：
本文研究了$w$无限带中单位直径$n$硬盘的位形空间$C(n,w)$的拓扑。我们描述的参数范围或“制度”，其中同源性$H_j [C(n,w)]$表现在定性不同的方式。证明了如果$w \ge j+2$，则同调$H_j[C(n, w)]$同构于平面上点的位形空间的同调$H_j[C(n, \mathbb{R}^2)]$ . $C(n, \mathbb{R}^2) $的Betti数由Arnold计算，因此作为同构的推论，$β_j[C(n,w)]$是$n$中次数为$2j$的多项式。另一方面，我们表明，如果$2 \le w \le j+1$，则$β_j [ C(n,w) ]$与$n$呈指数增长。我们的大部分工作是在仔细估计$β_j [ C(n,w) ]$在这一制度。我们还说明了，对于每个$n$，在$(w,j)$平面的同源“相图”-参数值，其中同源$H_j [C(n,w)]$是平凡的，非平凡的，同构与$H_j [C(n, \mathbb{R}^2)]$。出于硬球系统的相变的概念，我们讨论这些作为“同调固体，液体和气体”制度。
英文摘要：
We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $β_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $β_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $β_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.
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