Semi-coarse Spaces, Homotopy and Homology
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910623777947648 |
|---|---|
| author | Rieser, Antonio Treviño-Marroquín, Jonathan |
| author_facet | Rieser, Antonio Treviño-Marroquín, Jonathan |
| contents | We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02569 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Semi-coarse Spaces, Homotopy and Homology Rieser, Antonio Treviño-Marroquín, Jonathan Algebraic Topology General Topology Geometric Topology We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs. |
| title | Semi-coarse Spaces, Homotopy and Homology |
| topic | Algebraic Topology General Topology Geometric Topology |
| url | https://arxiv.org/abs/2210.02569 |