Geodesic
Homology of configuration spaces of hard squares in a rectangle
Alpert, Hannah1 ; Bauer, Ulrich2 ; Kahle, Matthew3 ; MacPherson, Robert4 ; Spendlove, Kelly5
1 Department of Mathematics and Statistics, Auburn University, Auburn, AL, United States
2 Department of Mathematics, Technical University of Munich, Munich, Germany
3 Department of Mathematics, Ohio State University, Columbus, OH, United States
4 School of Mathematics, Institute for Advanced Study, Princeton, NJ, United States
5 Mathematical Institute, University of Oxford, Oxford, United Kingdom
Algebraic and Geometric Topology, Tome 23 (2023) no. 6, pp. 2593-2626

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/agt.2023.23.2593
Keywords: homology, configuration spaces, statistical mechanics

Alpert, Hannah 1 ; Bauer, Ulrich 2 ; Kahle, Matthew 3 ; MacPherson, Robert 4 ; Spendlove, Kelly 5

1 Department of Mathematics and Statistics, Auburn University, Auburn, AL, United States
2 Department of Mathematics, Technical University of Munich, Munich, Germany
3 Department of Mathematics, Ohio State University, Columbus, OH, United States
4 School of Mathematics, Institute for Advanced Study, Princeton, NJ, United States
5 Mathematical Institute, University of Oxford, Oxford, United Kingdom 
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     title = {Homology of configuration spaces of hard squares in a rectangle},
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Alpert, Hannah; Bauer, Ulrich; Kahle, Matthew; MacPherson, Robert; Spendlove, Kelly. Homology of configuration spaces of hard squares in a rectangle. Algebraic and Geometric Topology, Tome 23 (2023) no. 6, pp. 2593-2626. doi: 10.2140/agt.2023.23.2593

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