Numerical invariants — Question 13.5

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 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. Write $\widetilde M$ for the universal cover of $M$ and $\widetilde{\mathcal F}$ for the full lifted foliation. The leaf space $L_{\mathcal F}=\widetilde M/\widetilde{\mathcal F}$ has its quotient topology, and can be non-Hausdorff. A taut foliation is $\mathbb R$-covered if this space is homeomorphic to $\mathbb R$. Its deck action is the holonomy representation $\rho_H:\pi_1(M)\longrightarrow\operatorname{Homeo}(L_{\mathcal F})$. Branching means nonseparated leaves in this space; one-sided branching occurs in only one transverse direction, and two-sided branching in both. For real cubical singular cycles $C=\sum_j a_j\sigma_j$ use $\sum_j|a_j|$. Define a foliated cubical analogue by restricting to cubes whose induced foliation has affine level-set form, and compare with the cubical norm using unrestricted representing cycles.

Statement recorded in the supplied source.

Develop the foliated Gromov norm using cubical chains and determine whether for every foliation $\mathcal F$ on a hyperbolic $3$-manifold its value on $[M]$ is strictly greater than the usual cubical Gromov norm of $[M]$. Include $\mathbb R$-covered foliations. Both sides must use the same cubical chain convention and normalization. [S2]

Short English statement

Define the cubical foliated norm and test strict inequality over the ordinary cubical fundamental-class norm, including real-covered foliations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300058 (AMR-102-0058). 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 13.5; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.