Surjectivity conjectures of Dupont and Monod

The conjectures ask whether bounded continuous cohomology surjects onto continuous cohomology for broad classes of semisimple Lie groups. Earlier work established important special cases, but the general question remained open in higher degrees.

Known results

• Dupont proved surjectivity in degree $2$ (classical result, date not given).
• Hermitian groups with finite centre and no compact factors: surjectivity in every degree (2009).
• The even-generator subring consists of bounded classes for general semisimple groups (2009).
• Full comparison-map isomorphism in degree $3$ for connected simple complex classical groups (2023).

September 2026 claimed proof

A September 2026 arXiv paper claims the surjectivity conjectures for connected semisimple Lie groups, using polylogarithmic cocycles representing Borel classes. This would settle the general family of comparison-map questions, but the claim is unrefereed and unverified.

Current status (as of September 2026): Special cases and low-degree results are established, while the general conjectures are claimed solved by an unrefereed preprint but remain unverified.