Bounded cohomology of transformation groups in all degrees
Abstract
Let M be a compact connected oriented smooth manifold, of dimension n≥2. Let 𝒯MG be one of the following transformation groups: Homeo0(M,μ), Diff0(M,μ), Symp0(M, ω) (in case M is symplectic) and $Ham(M, ω)$ (in case M is symplectic). Denote by H̅bd(𝒯M) reduced bounded cohomology of 𝒯M in degree d, and by EH̅d(𝒯M) reduced exact bounded cohomology of 𝒯M in degree d. In this paper we prove that if n=2 and M is a closed surface Σg or a disc 𝔻, then for every d≥ 2 dim(H̅bd(𝒯M))=∞. Moreover, dimEH̅d(𝒯M)=∞ if 𝒯M is either Diff0(Σg,μ) or Ham(Σg, ω), or Ham(𝔻, ω); or g>1. In case n>2, under certain conditions on π1(M), we prove that for every degree d≥ 2 dimEH̅d(𝒯M)=∞. In particular, these results hold for 𝒯M when M is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.
Related
- Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces
- Exact bounded cohomology of measure-preserving homeomorphism groups of nonorientable hyperbolic manifolds
- Fake quadrics with bounded cohomology property
- Smooth projective surfaces with bounded cohomology property II
- Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions
BibTeX
@misc{brandenbursky2026bounded,
title = {{Bounded cohomology of transformation groups in all degrees}},
author = {Michael Brandenbursky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11327},
eprint = {2610.11327},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11327},
}
RIS
TY - UNPB
TI - Bounded cohomology of transformation groups in all degrees
AU - Michael Brandenbursky
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11327
UR - https://arxiv.org/abs/2610.11327
AB - Let M be a compact connected oriented smooth manifold, of dimension n≥2. Let 𝒯_MG be one of the following transformation groups: Homeo_0(M,μ), Diff_0(M,μ), Symp_0(M, ω) (in case M is symplectic) and $Ham(M, ω)$ (in case M is symplectic). Denote by H̅_b^d(𝒯_M) reduced bounded cohomology of 𝒯_M in degree d, and by EH̅^d(𝒯_M) reduced exact bounded cohomology of 𝒯_M in degree d. In this paper we prove that if n=2 and M is a closed surface Σ_g or a disc 𝔻, then for every d≥ 2 dim(H̅b^d(𝒯_M))=∞. Moreover, dimEH̅^d(𝒯_M)=∞ if 𝒯_M is either Diff_0(Σ_g,μ) or Ham(Σ_g, ω), or Ham(𝔻, ω); or g>1. In case n>2, under certain conditions on π_1(M), we prove that for every degree d≥ 2 dimEH̅^d(𝒯_M)=∞. In particular, these results hold for 𝒯_M when M is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.
ER -