Homology of configuration spaces of hard squares in a rectangle

Abstract

We study the configuration spaces C(n; p, q) of n labeled 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 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 homologyvanishing 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.

Morse Theoretic Templates for High Dimensional Homology Computation

“…Proof of Theorem 1.3. As each X p is H n -embeddable for some n, it follows from Theorem 1.2 that for each p ∈ P, Mate(•, α p ) is an acyclic partial matching on X p , where α p is given in (1). It follows from Theorem 2.12 that w is an acyclic partial matching on X .…”

“…Note that Theorems 1.1-1.3 hold when the field K is replaced by a principal ideal domain R and the appropriate change is made to the definitions of incidence relation and partial matching. 1 However, working over fields both simplifies the exposition and allows for the use of iterative discrete Morse theory to compute both homology and Conley complexes, as in the algorithms Homology and ConnectionMatrix of Section 6.…”

“…. , α n ) where α i is defined as in (1). It follows from Proposition 4.1 that each α i is an acyclic partial matching.…”

“…, α d )) on a cell is O(2 d ) as in Theorem 1.1, and second, there are many more flowlines to follow in higher dimensions when computing ∂ Remark 6.1. We note that the templates scheme and the algorithm Homology implemented in the package PyCHomP have also been used to compute the homology of high dimensional complexes in [1].…”