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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 |
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Algebraic & Geometric Topology 23 (2023) 2593–2626
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Abstract |
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We study ordered configuration spaces of hard squares in a 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 , , , and the homology group 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. |
Keywords
homology, configuration spaces, statistical mechanics
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Mathematical Subject Classification
Primary: 55R80
Secondary: 57Q70, 82B26
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References |
Publication
Received: 20 November 2020
Revised: 4 October 2021
Accepted: 20 December 2021
Published: 7 September 2023
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Authors |
| Hannah Alpert | |
| Department of Mathematics and
Statistics Auburn University Auburn, AL United States |
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| Ulrich Bauer | |
| Department of Mathematics Technical University of Munich Munich Germany |
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| Matthew Kahle | |
| Department of Mathematics Ohio State University Columbus, OH United States |
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| Robert MacPherson | |
| School of Mathematics Institute for Advanced Study Princeton, NJ United States |
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| Kelly Spendlove | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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