Branched surfaces and triangulations — Question 7.5

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. Relative to a triangulation, a normal surface or lamination meets each tetrahedron in triangle and quadrilateral disk types; quadrilateral types must be compatible. The Haken sum for compatible normal surfaces is the cut-and-reglue operation along intersections that adds their normal coordinates. For a finite real singular chain $C=\sum_j a_j\sigma_j$, set $\|C\|_1=\sum_j|a_j|$ after equal simplices are combined. This is the chain norm relevant to the Gromov norm of homology. Normality relative to a singular cycle requires the pullback lamination in its simplices to have normal disk form.

Statement recorded in the supplied source.

Given an essential lamination $\Lambda$ of a $3$-manifold $M$ and a real singular cycle $C$ representing its fundamental class $[M]$, does there exist another representative $C'$ with $\|C'\|_1=\|C\|_1$ and an essential lamination $\Lambda'$ normal with respect to $C'$? For boundary cases use the relevant relative fundamental cycle. The question concerns normalization without increasing or decreasing the given chain norm. [S2]

Short English statement

Can a fundamental cycle be replaced with one of the same chain norm that is normal to some essential lamination?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300020 (AMR-102-0020). 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 7.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.