PreprintCC BY 4.0arXiv
The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group
Abstract
We prove that Hcb4(Sp(4,CC);RR)=0. Together with Blatz's secondary stability and the rank-one theorem of Bucher--Savini, this gives degree-four vanishing for all complex symplectic and odd complex orthogonal groups. In normalized symplectic Gram coordinates, we establish a bounded-primitive estimate and compute the measurable cohomology of the projective action. An explicit rational cocycle has divergent periods on a family of finite orbit cycles of uniformly bounded ℓ1-mass. A two-cone averaging construction extends the period estimate to measurable cochains and excludes bounded representatives of every nonzero degree-four measurable action class.
Related
- Relative symplectic cohomology in complex projective spaces
- Biran decomposition and relative symplectic cohomology
- Fake quadrics with bounded cohomology property
- Smooth projective surfaces with bounded cohomology property II
- Exact bounded cohomology of measure-preserving homeomorphism groups of nonorientable hyperbolic manifolds
BibTeX
@misc{gu2026fourth,
title = {{The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group}},
author = {Sixuan Gu and Yaoyu Cheng and Wei Qi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16848},
eprint = {2609.16848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16848},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group
AU - Sixuan Gu
AU - Yaoyu Cheng
AU - Wei Qi
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.16848
UR - https://arxiv.org/abs/2609.16848
AB - We prove that H_cb^4(Sp(4,CC);RR)=0. Together with Blatz's secondary stability and the rank-one theorem of Bucher--Savini, this gives degree-four vanishing for all complex symplectic and odd complex orthogonal groups. In normalized symplectic Gram coordinates, we establish a bounded-primitive estimate and compute the measurable cohomology of the projective action. An explicit rational cocycle has divergent periods on a family of finite orbit cycles of uniformly bounded ℓ^1-mass. A two-cone averaging construction extends the period estimate to measurable cochains and excludes bounded representatives of every nonzero degree-four measurable action class.
ER -