Branched surfaces and triangulations — Question 7.2

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 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 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.

Statement recorded in the supplied source.

Develop a hierarchy theory for branched surfaces: decompose a branched surface successively along train tracks into simpler pieces, and establish criteria for essential laminations carried by these pieces to glue to an essential lamination carried by the original branched surface. The choices of allowable decompositions, terminal pieces and gluing hypotheses are part of the requested theory. [S2]

Short English statement

Develop branched-surface hierarchies together with a gluing theorem for the essential laminations of their pieces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300017 (AMR-102-0017). 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 7.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.