Dowker's theorem for higher-order relations
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915339847073792 |
|---|---|
| author | de Silva, Vin Giusti, Chad Itskov, Vladimir Robinson, Michael Sazdanovic, Radmila Schonsheck, Nikolas Vaupel, Melvin Yoon, Iris H. R. |
| author_facet | de Silva, Vin Giusti, Chad Itskov, Vladimir Robinson, Michael Sazdanovic, Radmila Schonsheck, Nikolas Vaupel, Melvin Yoon, Iris H. R. |
| contents | Given a relation $R \subseteq I \times J$ between two sets, Dowker's Theorem (1952) states that the homology groups of two associated simplicial complexes, now known as Dowker complexes, are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to multiway relations of the form $R \subseteq I_1 \times I_2 \times \cdots \times I_m$. The theorem asserts functorial homotopy equivalences between $m$ multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker's Theorem. Our proof uses Smale's homotopy mapping theorem and factors through a cellular Dowker lemma that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called prod-complexes then transfer the results to simplicial complexes through a simplexification process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dowker's theorem for higher-order relations de Silva, Vin Giusti, Chad Itskov, Vladimir Robinson, Michael Sazdanovic, Radmila Schonsheck, Nikolas Vaupel, Melvin Yoon, Iris H. R. Algebraic Topology 55U10, 55N31 Given a relation $R \subseteq I \times J$ between two sets, Dowker's Theorem (1952) states that the homology groups of two associated simplicial complexes, now known as Dowker complexes, are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to multiway relations of the form $R \subseteq I_1 \times I_2 \times \cdots \times I_m$. The theorem asserts functorial homotopy equivalences between $m$ multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker's Theorem. Our proof uses Smale's homotopy mapping theorem and factors through a cellular Dowker lemma that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called prod-complexes then transfer the results to simplicial complexes through a simplexification process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them. |
| title | Dowker's theorem for higher-order relations |
| topic | Algebraic Topology 55U10, 55N31 |
| url | https://arxiv.org/abs/2506.10909 |