Foliations and Diffeomorphism Groups — Question 18

Definitions and mathematical statement

For a smooth bundle $\pi\colon E\to B$ with closed, connected, oriented surface fibre $\Sigma_g$ and orientation-preserving transition maps, let $T_\pi E$ be its oriented vertical tangent bundle. Its Miller--Morita--Mumford classes are \[ \kappa_i(\pi)=\pi_!\left(e(T_\pi E)^{i+1}\right)\in H^{2i}(B;{\mathbb{R}}), \] where $e$ is the real Euler class and $\pi_!$ is integration over the oriented surface fibre.

The source writes $G=\mathrm{Diff}(\Sigma_g)$. It does not specify the genus range, whether diffeomorphisms preserve orientation, or whether this group is given the discrete topology. Under the interpretation that $G$ is the discrete group of orientation-preserving diffeomorphisms, the preceding construction on the universal associated surface bundle gives $\kappa_i\in H^{2i}(G;{\mathbb{R}})$.

For a discrete group $G$, let $H_b^n(G;{\mathbb{R}})$ denote the cohomology of bounded group cochains with trivial real coefficients, and let \[ c_G\colon H_b^n(G;{\mathbb{R}})\longrightarrow H^n(G;{\mathbb{R}}) \] be the comparison map induced by inclusion into all group cochains. The source attributes to Morita, using Bott vanishing, the assertion that $\kappa_i=0$ in ordinary real cohomology for $i>2$.

Question, conditional on specifying the intended bounded classes. If $\kappa_i^b\in H_b^{2i}(G;{\mathbb{R}})$ denotes the bounded MMM class intended by the source, with $c_G(\kappa_i^b)=\kappa_i$, does \[ \kappa_i^b=0\quad\text{in }H_b^{2i}(G;{\mathbb{R}}) \qquad\text{for every integer }i>2 \] hold for the intended genera?

The supplied record does not construct or identify $\kappa_i^b$. The comparison identity alone does not specify such a class: once $\kappa_i=0$, the zero bounded class is already a lift. Thus merely asking whether a vanishing bounded lift exists would not capture the intended question. The intended bounded classes, genus quantifiers, and group conventions require clarification from the source.

Short English statement

The source asks whether the higher MMM classes, whose ordinary real-cohomology vanishing it attributes to Morita, also vanish as bounded-cohomology classes. A precise formulation requires identification of the intended bounded MMM classes and specification of the genus and diffeomorphism-group conventions.

Sources

• Local UnsolvedMath registry: permanent ID $800077$, exact source code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 18." The supplied registry labels the problem open; this status is not independently verified.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 18. Supplied source URL: https://arxiv.org/abs/2503.18274. The provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance claims, not an assertion of independent access or verification.
• The supplied statement attributes ordinary-cohomology vanishing for $i>2$ to Morita using the Bott vanishing theorem. No separate bibliographic reference for Morita's result is supplied.