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Homology of configuration spaces of hard squares in a rectangle

Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson and Kelly Spendlove

Algebraic & Geometric Topology 23 (2023) 2593–2626

We study ordered configuration spaces C(n;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 Hj[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.

homology, configuration spaces, statistical mechanics
Mathematical Subject Classification
Primary: 55R80
Secondary: 57Q70, 82B26
Received: 20 November 2020
Revised: 4 October 2021
Accepted: 20 December 2021
Published: 7 September 2023
Hannah Alpert
Department of Mathematics and Statistics
Auburn University
Auburn, AL
United States
Ulrich Bauer
Department of Mathematics
Technical University of Munich
Matthew Kahle
Department of Mathematics
Ohio State University
Columbus, OH
United States
Robert MacPherson
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Kelly Spendlove
Mathematical Institute
University of Oxford
United Kingdom

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