The 15 Puzzle and homological stability in the space direction
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910032570875904 |
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| author | González, Jesús Kahle, Matthew Wawrykow, Nicholas |
| author_facet | González, Jesús Kahle, Matthew Wawrykow, Nicholas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21300 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 15 Puzzle and homological stability in the space direction González, Jesús Kahle, Matthew Wawrykow, Nicholas Algebraic Topology Geometric Topology 55R80, 52C15, 82B26 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. |
| title | The 15 Puzzle and homological stability in the space direction |
| topic | Algebraic Topology Geometric Topology 55R80, 52C15, 82B26 |
| url | https://arxiv.org/abs/2602.21300 |