Homology of configuration spaces of hard squares in a rectangle

Hannah Alpert; Ulrich Bauer; Matthew Kahle; Robert MacPherson; Kelly Spendlove

doi:10.2140/agt.2023.23.2593

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arXiv · 2010.14480

Homology of configuration spaces of hard squares in a rectangle

Hannah Alpert·Ulrich Bauer·Matthew Kahle·Robert MacPherson·Kelly Spendlove

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Abstract

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.

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Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, Kelly Spendlove. 2020-10-27. Homology of configuration spaces of hard squares in a rectangle. https://doi.org/10.2140/agt.2023.23.2593

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