Boundedness of Tautological Kappa Classes

Definitions and mathematical statement

The supplied record contains a principal boundedness question and two additional questions in its original statement. All three are retained below; several parameters needed for a definitive formulation are unspecified.

For a conventional interpretation of the principal question, let $\Sigma_g$ be a closed, connected, oriented surface of genus $g\geq 2$, and let $\operatorname{Mod}_g=\pi_0(\operatorname{Diff}^{+}(\Sigma_g))$ be its orientation-preserving mapping class group. In the standard classifying-space model, write $\pi:E_g\to B\operatorname{Mod}_g$ for the universal oriented surface bundle and $e(T_\pi)\in H^2(E_g;{\mathbb{R}})$ for the Euler class of its vertical tangent bundle. Define the real tautological, or Miller--Morita--Mumford, classes by \[ \kappa_i=\pi_!\left(e(T_\pi)^{i+1}\right)\in H^{2i}(\operatorname{Mod}_g;{\mathbb{R}}),\qquad i\geq 0, \] where $\pi_!$ denotes integration over the oriented surface fibre. For a discrete group $G$, let $H_b^n(G;{\mathbb{R}})$ denote the cohomology of the bounded real group cochains, with trivial coefficient action. Inclusion into all group cochains induces the comparison map \[ c_G^n:H_b^n(G;{\mathbb{R}})\longrightarrow H^n(G;{\mathbb{R}}). \] An ordinary real cohomology class is called bounded if it lies in the image of this map.

1. For the tautological classes intended by the source, does \[ \kappa_i\in\operatorname{im}\left(c_{\operatorname{Mod}_g}^{2i}\right) \] hold? The closed-surface convention, real coefficients, and the displayed range $g\geq 2$ provide a standard precise setting, but the supplied record does not explicitly specify the surface conventions, coefficient ring, genus range, or indices intended by its question. Consequently, no universal quantification over all displayed pairs $(g,i)$ is asserted to have been verified from the source.

2. For a torsion-free subgroup $\Gamma\leq\operatorname{Mod}_g$ of finite index, is \[ H^{2g-1}\left(\Gamma;H^{2g-2}(C_g)\right) \] infinitely generated? This expression is retained exactly at the level supplied: the object $C_g$, the coefficients of its cohomology, and the action of $\Gamma$ on $H^{2g-2}(C_g)$ are not defined in the record. Thus the coefficient module and the resulting group cohomology are not yet determined. The genus range, whether the question concerns every such $\Gamma$, and whether infinite generation means as an abelian group or over a specified coefficient ring also require clarification.

3. Does there exist a finite-volume hyperbolic $3$-manifold $M$ and an injective group homomorphism \[ \rho:\pi_1(M)\hookrightarrow\operatorname{SL}_3({\mathbb{Z}}), \qquad \operatorname{SL}_3({\mathbb{Z}})=\{A\in M_3({\mathbb{Z}}):\det A=1\}? \] Here finite volume refers to the hyperbolic metric. The record does not state whether completeness or orientability is required, or whether $M$ must be closed or may be noncompact. These conventions remain unresolved rather than being imposed on the existential question.

The supplied dated assessment labels the record partially solved and states that odd kappa classes are bounded, that stable MMM polynomials are unbounded, and that the coefficient-cohomology and $\operatorname{SL}_3({\mathbb{Z}})$ embedding questions remain unresolved. These are supplied curation claims, not independently verified mathematical conclusions. The index and genus ranges of the first claim, and the quantifiers and meaning of stability in the polynomial claim, are not specified sufficiently to turn them into additional precise assertions.

Short English statement

Ask whether the intended tautological kappa classes have bounded cohomology representatives. The original record also asks about infinite generation of a group cohomology group with unspecified coefficients and about embedding the fundamental group of a finite-volume hyperbolic $3$-manifold into $\operatorname{SL}_3({\mathbb{Z}})$. Missing conventions and the supplied dated status claims require human review.

Sources

• Local UnsolvedMath registry, permanent ID $30003297$, exact source problem code catalogue entry, title Boundedness of Tautological Kappa Classes; supplied complete source record from the Oberwolfach Reports Open Problems collection. The project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; neither file was independently inspected for this draft.
• Supplied source citation: Surface Bundles (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/56, pp. 3149--3195. https://doi.org/10.4171/owr/2016/56. The literature-source list labels this reference "OWR 56 (2017)." The record assigns proposed year 2016 and supplies no named proposer.
• Supplied literature reference: Kastenholz, "bounded cohomology stability." https://arxiv.org/abs/2308.14171. No fuller bibliographic citation is supplied in the record.
• Literature checked at: 2026-08-21. This exact date belongs to supplied dated curation, not an independently verified assessment. That curation labels the problem partially solved and reports boundedness of odd kappa classes, unboundedness of stable MMM polynomials, and unresolved coefficient-cohomology and $\operatorname{SL}_3({\mathbb{Z}})$ embedding questions. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified, with a supplied assertion that the clean statement was checked against the official report. This draft has not independently visited or checked either reference. The clean statement retains only the boundedness question, whereas the original statement contains all three questions reproduced above.