Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.10976 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917945861472256 |
|---|---|
| author | Dochtermann, Anton Matsushita, Takahiro |
| author_facet | Dochtermann, Anton Matsushita, Takahiro |
| contents | In his work on molecular spaces, Ivashchenko introduced the notion of an $\mathfrak{I}$-contractible transformation on a graph $G$, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the $\mathfrak{I}$-homotopy type of a graph, and showed that $\mathfrak{I}$-contractible transformations preserve the simple homotopy type of $C(G)$, the clique complex of $G$. In other work, Boulet, Fieux, and Jouve introduced the notion of $s$-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that $s$-homotopy preserves $\mathfrak{I}$-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs $G$ and $H$ are $\mathfrak{I}$-homotopy equivalent if and only if $C(G)$ and $C(H)$ are simple homotopy equivalent. We also show that a finite graph $G$ is $\mathfrak{I}$-contractible if and only if $C(G)$ is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10976 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Simple homotopy of flag simplicial complexes and contractible contractions of graphs Dochtermann, Anton Matsushita, Takahiro Combinatorics 57Q10, 04E45 In his work on molecular spaces, Ivashchenko introduced the notion of an $\mathfrak{I}$-contractible transformation on a graph $G$, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the $\mathfrak{I}$-homotopy type of a graph, and showed that $\mathfrak{I}$-contractible transformations preserve the simple homotopy type of $C(G)$, the clique complex of $G$. In other work, Boulet, Fieux, and Jouve introduced the notion of $s$-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that $s$-homotopy preserves $\mathfrak{I}$-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs $G$ and $H$ are $\mathfrak{I}$-homotopy equivalent if and only if $C(G)$ and $C(H)$ are simple homotopy equivalent. We also show that a finite graph $G$ is $\mathfrak{I}$-contractible if and only if $C(G)$ is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices. |
| title | Simple homotopy of flag simplicial complexes and contractible contractions of graphs |
| topic | Combinatorics 57Q10, 04E45 |
| url | https://arxiv.org/abs/2312.10976 |