# Configuration spaces of hard squares in a rectangle

• Published in 2020
In the collection
We study the configuration spaces C(n;p,q) of n labeled hard squares in a p by 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.

### BibTeX entry

@article{Configurationspacesofhardsquaresinarectangle,
title = {Configuration spaces of hard squares in a rectangle},
abstract = {We study the configuration spaces C(n;p,q) of n labeled hard squares in a p
by 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.},
url = {http://arxiv.org/abs/2010.14480v1 http://arxiv.org/pdf/2010.14480v1},
year = 2020,
author = {Hannah Alpert and Ulrich Bauer and Matthew Kahle and Robert MacPherson and Kelly Spendlove},
comment = {},
urldate = {2020-11-06},
archivePrefix = {arXiv},
eprint = {2010.14480},
primaryClass = {math.AT},
collections = {puzzles}
}