Homology of configuration spaces of hard squares in a rectangle
Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, Kelly Spendlove · Algebraic & Geometric Topology · 2023
We study ordered configuration spaces C.nI p; q/ of n hard squares in a p q rectangle, a generalization of the well-known "15 puzzle".Our main interest is in the topology of these spaces.Our first result describes a cubical cell complex and proves that it is homotopy equivalent to the configuration space.We then focus on determining for which n, j , p, and q the homology group H j ŒC.nI 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.