Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.04745 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909379832315904 |
|---|---|
| author | Margolis, Alexander |
| author_facet | Margolis, Alexander |
| contents | Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological dimension of G as a metric space coincides with the cohomological dimension of $G$ as a group whenever the latter is finite. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or $\infty$. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and $H^k(G,FG)=0$ for k<n, then $\dim H^n(G,FG)$=0,1 or $\infty$, significantly extending a result of Farrell from 1975. Moreover, in the case $\dim H^n(G,FG)=1$, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coarse homological invariants of metric spaces Margolis, Alexander Group Theory Metric Geometry 20J05, 20J06, 20F65, 51F30 Inspired by group cohomology, we define several coarse topological invariants of metric spaces. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group, then the coarse cohomological dimension of G as a metric space coincides with the cohomological dimension of $G$ as a group whenever the latter is finite. Extending a result of Sauer, it is shown that coarse cohomological dimension is monotone under coarse embeddings, and hence is invariant under coarse equivalence. We characterise unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one. A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or $\infty$. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and $H^k(G,FG)=0$ for k<n, then $\dim H^n(G,FG)$=0,1 or $\infty$, significantly extending a result of Farrell from 1975. Moreover, in the case $\dim H^n(G,FG)=1$, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces. |
| title | Coarse homological invariants of metric spaces |
| topic | Group Theory Metric Geometry 20J05, 20J06, 20F65, 51F30 |
| url | https://arxiv.org/abs/2411.04745 |