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arXiv · 2609.16848

The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group

Sixuan Gu·Yaoyu Cheng·Wei Qi
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Abstract

We prove that $H_{\mathrm{cb}}^4(\Sp(4,\CC);\RR)=0$. 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 $\ell^1$-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.

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Sixuan Gu, Yaoyu Cheng, Wei Qi. 2026-09-17. The Fourth Continuous Bounded Cohomology of the Complex Symplectic Group. https://arxiv.org/abs/2609.16848

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