Sublaminations and superlaminations — Question 6.5

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A surface lamination $\Lambda$ is a closed subset of $M$ locally homeomorphic to a product of a disk with a closed transverse set. It is essential when it has no sphere leaf or torus leaf bounding a solid torus, and its complementary regions are irreducible, have incompressible boundary, and have no compressing monogon. Such a monogon is a compressing disk with one boundary arc on a boundary leaf and the other along a transverse interval fiber. It is genuine if some complementary region is not an interval bundle. A sublamination is a closed union of leaves. Write $\widetilde M$ for the universal cover of $M$ and $\widetilde\Lambda$ for the full inverse-image lamination under the covering map. A loosesse lamination obeys the essential-lamination conditions except that compressing monogons are allowed. A leaf is properly embedded if its inclusion is an embedding and the inverse image of every compact ambient set is compact.

Statement recorded in the supplied source.

Determine useful topological consequences of a loosesse lamination. In particular, must every leaf of $\widetilde\Lambda$ be properly embedded in $\widetilde M$? Does existence of such a lamination imply $\widetilde M\cong\mathbb R^3$? The broader utility question is part of the task, in addition to these two explicit tests. [S2]

Short English statement

Determine the usefulness of loosesse laminations, whether their lifted leaves are proper, and whether they force the universal cover to be Euclidean three-space.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300014 (AMR-102-0014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 6.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.