Classical 3-manifold theory — Question 9.1

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. Dehn surgery on a link removes disjoint solid-torus neighborhoods and reglues solid tori with chosen meridian slopes. A transverse link has every component everywhere transverse to the given foliation.

Statement recorded in the supplied source.

Does there exist a fixed $3$-manifold $M$ such that, for every tautly foliated $3$-manifold $(N,\mathcal F)$, some link $L\subset N$ transverse to $\mathcal F$ has surgery yielding $M$? All surgery slopes and the link may depend on $(N,\mathcal F)$; $M$ must be independent of them. [S2]

Short English statement

Can surgery on a link transverse to any given taut foliation always produce one fixed universal manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300034 (AMR-102-0034). 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 9.1; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.