Immersed objects — Question 14.4

Definitions and mathematical statement

A codimension-one lamination of a three-manifold is locally modeled on $D^2\times K$, where $K$ is a closed subset of an interval, with compatible two-dimensional plaques forming leaves. In the standard embedded essential-lamination setting, genuine means that the complementary structure has a component beyond the interval-bundle case, often described by nonempty guts after separating interval-bundle regions.

Are there interesting examples of the source's "total genuine laminations"? Construct examples satisfying that intended definition and explain the features making them nontrivial. The word total, the allowed ambient manifolds, and the intended notion of interesting are not defined in the supplied item. Its placement under "Immersed objects" also leaves unclear whether the objects are embedded laminations or immersed analogues.

The research summary proposes that total means every complementary region is an interval bundle. That would conflict with genuineness under the usual embedded convention above, so it cannot be adopted without checking the missing source definition. No alternative meaning of total or corrected complement condition is invented. The record therefore does not determine a precise example-existence class by itself.

Short English statement

Construct nontrivial examples of total genuine laminations in the source's sense. Total and the immersed/embedded category are undefined, and the summary's interval-bundle interpretation conflicts with the usual genuine-lamination convention.

Sources

• Calegari, Problems in foliations and laminations of 3-manifolds (2002), Question 14.4, "Immersed objects," https://arxiv.org/abs/math/0209081; accessed 2026-07-29, source-TeX extraction. The provenance inventories 67 numbered questions.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10300064$, source code catalogue entry, Immersed objects --- Question 14.4. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.