The 15 Puzzle and homological stability in the space direction

TL;DR

This paper investigates the homological stability of ordered configuration spaces of squares in rectangles, establishing bounds on rectangle dimensions that preserve homology and connecting it to classical configuration spaces.

Contribution

It sharpens the stable range for homological stability of square configuration spaces and relates it to the classical configuration space of points in the plane.

Findings

Homological stability holds for large enough rectangles.

Bounds on rectangle dimensions are provided in terms of n and k.

Most rectangles can be almost filled with squares without losing homological information.

Abstract

The ordered configuration space of $n$ open unit squares in the $w$ by $h$ rectangle exhibits homological stability in the space direction. That is, for fixed $n$ and fixed homological degree $k$, once the underlying rectangle is large enough, making it any larger does not change the $k$-th homology of the square configuration space. In this paper, we sharpen the stable range. Finding bounds for $w$ and $h$ in terms of $n$ and $k$, we prove that most rectangles can be almost entirely filled with squares and there still be an isomorphism between the $k$-th homology of the resulting square configuration space and the $k$-th homology of the ordered configuration space of $n$ points in the plane.