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The 15 Puzzle and homological stability in the space direction

Jesús González, Matthew Kahle, Nicholas Wawrykow

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.

The 15 Puzzle and homological stability in the space direction

Abstract

The ordered configuration space of open unit squares in the by rectangle exhibits homological stability in the space direction. That is, for fixed and fixed homological degree , once the underlying rectangle is large enough, making it any larger does not change the -th homology of the square configuration space. In this paper, we sharpen the stable range. Finding bounds for and in terms of and , we prove that most rectangles can be almost entirely filled with squares and there still be an isomorphism between the -th homology of the resulting square configuration space and the -th homology of the ordered configuration space of points in the plane.
Paper Structure (15 sections, 42 theorems, 204 equations, 13 figures)

This paper contains 15 sections, 42 theorems, 204 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Stability Patterns in Configuration Spaces
  3. Disk Configuration Spaces
  4. Outline
  5. Acknowledgments
  6. Square Configuration Space
  7. Big Square Configuration Space
  8. Our Mayer--Vietoris Spectral Sequences
  9. The Mayer--Vietoris Spectral Sequence: An Overview
  10. Covering $SF_{n}(R_{w,h})$ and $F_{n}(\mathbb{R}^{2})$
  11. The Complex of Injective Words
  12. The Arc Resolution Spectral Sequence
  13. The $d^{r}$-Differentials of Our Augmented Mayer--Vietoris Spectral Sequences
  14. Proof of Theorem \ref{['space homological stability']}
  15. A Tighter Bound for $k=1$

Key Result

Theorem 1.3

If $w\ge h\ge k+2$ and $wh-n\ge \max\{(k+1)(k+2), hk+2\}$, then the inclusion of $SF_{n}(R_{w,h})$ into $F_{n}(\mathbb{R}^{2})$ induces an isomorphism

Figures (13)

  • Figure 1: The initial and target states of the 15 Puzzle.
  • Figure 2: The first Betti number of $SF_{8}(R_{w,h})$. If $w,h\ge 3$ and $wh-8\ge 6$, then Theorem \ref{['almost sharp homological stability for k=1']} tells us that $H_{1}(SF_{8}(R_{w,h}))$ stabilizes to $H_{1}(F_{8}(\mathbb{R}^{2}))\cong \mathbb{Z}^{28}$. Results of Alpert and Manin alpert2021configuration1 prove that if $w\ge 8$, then $H_{1}(SF_{8}(R_{w,2}))\cong H_{1}(SF_{8}(\mathbb{S}_{2}))\cong \mathbb{Z}^{1218}$, where $SF_{8}(\mathbb{S}_{2})$ is the ordered configuration space of $8$ open unit squares in the infinite strip of height $2$.
  • Figure 3: The wheel $W(1,2,\dots, n)\in H_{n-1}(F_{n}(\mathbb{R}^{2}))$.
  • Figure 4: The wheel $W(1,2,\dots, n)\in H_{n-1}(SF_{n}(R_{n,n}))$. Squares $1$ and $2$ orbit each other in a $2\times 2$ square, square $3$ orbits this big square in a $3\times 3$ square, etc. As a class in $H_{n-1}(F_{n}(\mathbb{R}^{2}))$, this class is depicted in Figure \ref{['W(123n)onpoints']}.
  • Figure 5: Two non-homologous classes in $H_{1}(SF_{3}(R_{3,2}))$. Despite the fact these cycles can be viewed as products of the same set of wheels, they represent different classes in homology, as there is no way to move the wheel $W(3)$ past the wheel $W(1,2)$, which needs the entire height of the rectangle.
  • ...and 8 more figures

Theorems & Definitions (74)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 64 more