Kirby Problem 4.108

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 four-dimensional Liouville domain $(W,\lambda)$ is a compact smooth four-manifold with boundary, $d\lambda$ symplectic, and outward transverse Liouville vector field $V$ defined by $\iota_Vd\lambda=\lambda$. Its boundary contact structure is $\ker(\lambda|_{\partial W})$. A Weinstein domain also has a Morse function for which $V$ is gradient-like. A coorientation-preserving contactomorphism identifies the boundary contact hyperplane fields.

Given two Liouville domains $(W_+,\lambda_+)$ and $(W_-,\lambda_-)$ and a contactomorphism $\Phi:\partial W_-\to\partial W_+$, the convex-hypersurface gluing construction produces an $\mathbb R$-invariant cooriented contact structure $\xi$ on $\mathbb R\times X$, where \[ X=W_+\cup_\Phi(-W_-). \] The minus sign means reversal of the symplectic orientation on the negative region. In the standard invariant model the contact form is $f\,dt+\beta$ on $\mathbb R\times X$; its dividing set is $\Gamma=f^{-1}(0)$, and the positive/negative regions have Liouville forms $\beta_\pm=\pm\beta/f$. This specifies the usual gluing germ; the two original domains give its compact positive and negative pieces. The hypersurface $\{0\}\times X$ is convex: a vector field preserving the contact structure is transverse to it. [S3]

A five-dimensional contact structure is overtwisted if it contains an embedded Borman--Eliashberg--Murphy overtwisted four-disk with the standard overtwisted contact germ, as in the source's definition. It is tight if it is not overtwisted.

Statement recorded in the supplied source.

Retain all three questions:

1. (a) When both $W_+$ and $W_-$ are Weinstein, find a necessary and sufficient topological criterion on $(\Phi,W_+,W_-)$ for the resulting $\xi$ to be overtwisted; this is the proposed four-dimensional analogue of Giroux's surface criterion.
2. (b) Does every four-manifold $X$ admit such a decomposition with tight associated contact structure? A decomposition by the specified compact Liouville domains necessarily gives a closed orientable $X$; thus the substantive universal version is for closed oriented four-manifolds. If the word "every" is read to include noncompact or nonorientable manifolds, those cannot admit this kind of decomposition.
3. (c) Is there a Liouville domain $W$ and a boundary contactomorphism $\Phi:\partial W\to\partial W$ extending to a diffeomorphism $\Psi:W\to W$, for which the associated contact structure on $\mathbb R\times(W\cup_\Phi(-W))$ is overtwisted?

Part (c) only asks that $\Phi$ extend smoothly, and does not assume a symplectic extension or that $W$ is Weinstein. [S2]

Short English statement

Seek a topological overtwistedness criterion for glued Weinstein pieces, tight Liouville decompositions of four-manifolds, and an overtwisted double produced by a boundary contactomorphism that extends smoothly.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2984 (KP-4.108). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.108 and Remarks, PDF page 280. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Russell Avdek, An algebraic generalization of Giroux’s criterion, arXiv:2307.09068v3, Sections 3.3 and 4.3, pages 21 and 40--41. https://arxiv.org/pdf/2307.09068v3.