Volume and Euler classes in bounded cohomology of transformation groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913677000572928 |
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| author | Brandenbursky, Michael Marcinkowski, Michał |
| author_facet | Brandenbursky, Michael Marcinkowski, Michał |
| contents | Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,ω)$ the group of measure preserving homeomorphisms of $M$, where $ω$ is a finite measure induced by a volume form. In this paper we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0(M,ω)$ and $\operatorname{Homeo}(M,ω)$ respectively, and in several cases prove their non-triviality. More precisely, we define:
- Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,ω))$ where $M$ is a hyperbolic manifold of dimension $n$.
- Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,ω))$ where $S$ is a closed hyperbolic surface.
We show that Euler classes have positive norms for any closed hyperbolic $S$ and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds, and hence they are non-trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15974 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Volume and Euler classes in bounded cohomology of transformation groups Brandenbursky, Michael Marcinkowski, Michał Geometric Topology Dynamical Systems Group Theory 51, 53, 55, 57K10, 58D05, 37E30 Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,ω)$ the group of measure preserving homeomorphisms of $M$, where $ω$ is a finite measure induced by a volume form. In this paper we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0(M,ω)$ and $\operatorname{Homeo}(M,ω)$ respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,ω))$ where $M$ is a hyperbolic manifold of dimension $n$. - Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,ω))$ where $S$ is a closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic $S$ and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds, and hence they are non-trivial. |
| title | Volume and Euler classes in bounded cohomology of transformation groups |
| topic | Geometric Topology Dynamical Systems Group Theory 51, 53, 55, 57K10, 58D05, 37E30 |
| url | https://arxiv.org/abs/2308.15974 |