Homology of configuration spaces of hard squares in a rectangle

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.

Final version, to appear in "Algebraic & Geometric Topology"

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