Classical 3-manifold theory — Question 9.2

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 codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct.

Statement recorded in the supplied source.

Find explicit base foliated manifolds and fundamental operations on foliations from which every tautly foliated pair $(M,\mathcal F)$ can be constructed by repeated applications. Specify both the starting family and the allowed operations, and prove their completeness; merely generating the underlying manifolds without their foliations is insufficient. [S2]

Short English statement

Give a complete construction calculus for taut foliations, with explicit starting examples and elementary operations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300035 (AMR-102-0035). 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 9.2; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.