Coarse foliations — Question 12.2

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 cone field on a smooth $3$-manifold $M$ is a smooth symmetric covariant $2$-tensor $C$ of signature $(2,1)$. An orientation of $C$ is a continuous choice at every $p\in M$ of one of the two components of $\{v\in T_pM:C_p(v,v)<0\}$, called the positive timelike cone. A positively timelike path has its tangent in that cone. The cone field is recurrent if every ordered pair of points of $M$ is joined by such a path. It is homotopically essential if every closed positively timelike path is noncontractible in $M$. A taut cone field is an oriented cone field with both properties. These are the conventions of Definition 2.4 in [S3]. A supported embedded loop is an embedded closed positively timelike path. For such a loop $K$, Dehn surgery removes a tubular neighborhood of $K$ and attaches a solid torus whose meridian is identified with a slope, an unoriented primitive curve class on the boundary torus. Nontrivial surgery means a slope different from the original meridian.

Statement recorded in the supplied source.

Does every hyperbolic $3$-manifold admit a taut cone field? For a taut cone field and a supported embedded loop, can the cone field remain taut after surgery at every slope outside a finite set? Is there a class of examples in which it remains taut after every nontrivial surgery? The surgery questions include constructing a suitable extension of the cone field over the filled solid torus while retaining it away from the surgery neighborhood. [S2]

Short English statement

Does every hyperbolic three-manifold admit a recurrent cone field with only noncontractible supported loops, and how broadly can tautness survive surgery on a supported loop?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300052 (AMR-102-0052). 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 12.2; Definitions 1.1--1.10 and local definitions and Remark (2). https://arxiv.org/abs/math/0209081v1.
• S3 Danny Calegari, Useful branched surfaces which carry nothing, version 0.82, arXiv:math/0010146v1 (2000), Definition 2.4, pp. 5--6 (withdrawn preprint, cited here for its cone-field terminology). https://arxiv.org/abs/math/0010146v1.