论文标题
矩形中硬正方形的配置空间的同源
Homology of configuration spaces of hard squares in a rectangle
论文作者
论文摘要
我们研究订购的配置空间$ c（n; p，q）$ n $ hard Squares的$ p \ times q $矩形，这是众所周知的“ 15个拼图”的概括。我们的主要兴趣是这些空间的拓扑。我们的第一个结果是描述一个立方细胞复合物，并证明其同等的配置空间。然后，我们专注于确定哪种$ n $，$ j $，$ p $和$ q $同源组$ h_j [c（n; p，q）] $是不繁琐的。我们证明了基于细胞复合物的离散摩尔斯理论，证明了三个同源性变化定理。然后，我们描述了几个明确的非平凡循环系列，以及一种在参数之间插值的方法，以填写“大规模”非平凡同源性的大部分图片。
We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p \times q$ rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.
