Classical 3-manifold theory — Question 9.5

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. A Heegaard splitting $M=H_1\cup_\Sigma H_2$ has distance equal to the distance in the curve graph of $\Sigma$ between the meridian-disk boundary sets of $H_1,H_2$. The curve graph has essential simple closed curves as vertices and disjointness edges.

Statement recorded in the supplied source.

Determine what useful constraints or filtration structure Heegaard-splitting distance imposes on essential laminations in $M$. In particular, if $M$ has a Heegaard splitting of distance at least $2g$, what conclusions can be drawn about its essential laminations? Compare the surface theorem that an essential surface of genus $g$ bounds splitting distance above by $2g$, while seeking genuinely lamination-theoretic consequences. [S2]

Short English statement

Determine how large Heegaard distance constrains essential laminations, extending the familiar distance bound from incompressible surfaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300038 (AMR-102-0038). 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 9.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.