Problem 7.20 — (S.

Definitions and mathematical statement

The source asks to specialize its omitted Problem 7.19 to bundles obtained from ordinary cohomology, and for real additive bundles to compare quantum hyperbolic invariants (QHI) with the Thurston norm. Ordinary $H^1(M;\mathbb R)$ classes can be viewed as additive characters of $\pi_1(M)$, but the embedding of these characters into the bundle category used by QHI is not supplied.

For a compact oriented three-manifold in the usual Thurston-norm setting, an integral cohomology class $a$ has norm/seminorm \[ x_M(a)=\min_{S\text{ represents the dual of }a} \sum_{S_i\text{ a component of }S}\max\{0,-\chi(S_i)\}, \] extended to real classes by homogeneity and convexity. The exact boundary/duality convention depends on the manifold class. The source asks whether the intended QHI, with remaining data fixed, is constant on faces of the corresponding unit sphere $x_M(a)=1$.

Writing only placeholders $I_r(M,a)$ for a chosen QHI level and cohomology-induced bundle, the constancy target would be \[ a,b\text{ in the same intended face}\quad\Longrightarrow\quad I_r(M,a)=I_r(M,b). \] The excerpt does not specify relative interiors versus closed faces or cones on faces, the treatment of a seminorm kernel, quantum levels, phase ambiguities, or auxiliary link/normalization data. None is chosen silently, and the full requested specialization cannot be reconstructed without Problem 7.19 and the preceding bundle construction.

The summary only calls the question not obviously resolved. Its open/triage label is qualified source information without an exact current resolution reference.

Short English statement

Study QHI for bundles induced by ordinary cohomology, especially real additive classes, and ask whether they are constant along the intended faces of the Thurston-norm unit sphere. The preceding problem, bundle construction and face/equality conventions are missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400135$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 7.20, attributed to S. Baseilhac and R. Benedetti, PDF page 114, referring to Problem 7.19: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29.
• The preceding QHI/bundle definitions and full Problem 7.19 are not supplied; no separate resolution reference is included.