Kirby Problem 5.17

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.

Let $(W,\omega)$ be an exact symplectic manifold. A Liouville vector field $V$ satisfies $\mathcal L_V\omega=\omega$, equivalently $d(\iota_V\omega)=\omega$. A Weinstein structure $(\omega,V,\phi)$ additionally has a Morse Lyapunov function $\phi$: $V$ is gradient-like for $\phi$ and $d\phi(V)>0$ away from its critical points, with the usual local gradient condition at them. On a compact domain $V$ is outward transverse to the boundary; on an open Weinstein manifold the exhaustion and completeness conditions are imposed. A Weinstein homotopy is a one-parameter family of such structures allowing isolated ordinary birth/death critical-point degeneracies. In the fixed-form question below $\omega$ is kept fixed and the family remains convex at the boundary (or at infinity).

Statement recorded in the supplied source.

Given any two Weinstein structures $(W,\omega,V_1,\phi_1)$ and $(W,\omega,V_2,\phi_2)$ on the same symplectic manifold, does there exist a Weinstein homotopy joining them? No flexibility, common Stein complex structure, or simple-connectivity hypothesis is assumed. [S2]

Short English statement

Can any two Weinstein structures for one fixed symplectic form be joined through Weinstein structures?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3024 (KP-5.17). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 5.17, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.