Miscellaneous — Question 15.1

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 branched surface $B\subset M$ is a smoothly tangent $2$-complex with the standard local branching models. Its fibered neighborhood $N(B)$ is an interval bundle with the usual identifications over the branch locus. A lamination is carried by $B$ if it lies in $N(B)$ transverse to interval fibers, and fully carried if it meets every fiber. Splitting means cutting this fibered neighborhood along carried surface pieces and taking the induced branched surface. A coorientation orders the points of a transverse interval. A transverse set is well-ordered when every nonempty subset has a least point in that order. An ordinal is the order type of a well-ordered set; ordinary branched-surface weights use numerical weights satisfying matching rules along branches.

Statement recorded in the supplied source.

Describe which cooriented surface laminations in $3$-manifolds can have well-ordered transverse spaces, and the possible order types and topological consequences. Develop a useful theory of branched surfaces with ordinal-valued weights, specifying appropriate matching and realization rules instead of assuming real addition applies unchanged to ordinals. [S2]

Short English statement

Classify laminations with well-ordered transversals and develop a meaningful theory of ordinal weights on branched surfaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300066 (AMR-102-0066). 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 15.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.