Foliations and Diffeomorphism Groups — Question 7

Definitions and mathematical statement

The supplied record does not specify the class of manifolds $M$, the contact invariant $c$, or its pairing. The following definitions explain the usual three-dimensional terminology conditionally; they do not resolve these omissions.

If $M$ is an oriented smooth three-dimensional manifold, a cooriented contact structure is a plane field $\xi=\ker\alpha$, where $\alpha$ is a globally defined one-form and $\alpha\wedge{\,\mathrm{d}}\alpha$ is nowhere zero. It is positive or negative according to whether $\alpha\wedge{\,\mathrm{d}}\alpha$ agrees or disagrees with the orientation of $M$. The source describes the structures as "(co)oriented" without specifying the intended orientation data. In this three-dimensional setting, a foliation by surfaces is Reebless if it contains no Reeb component, meaning an embedded solid torus with the standard Reeb foliation, whose boundary is a torus leaf and whose interior leaves are planes.

With the manifold hypotheses and invariant conventions left as intended by the source, let $(\xi_-,\xi_+)$ be a pair of (co)oriented contact structures of opposite signs on $M$. Assume \[ \langle c(\xi_-),c(\xi_+)\rangle=1. \] Must $M$ admit a Reebless foliation? Equivalently, the requested implication is \[ \langle c(\xi_-),c(\xi_+)\rangle=1 \quad\Longrightarrow\quad \text{there exists a Reebless foliation on }M. \] Here the groups containing the contact invariants, the pairing, and the coefficient system in which $1$ is interpreted remain unspecified. The source also leaves the dimension, compactness, boundary conditions, and orientation hypotheses on $M$, and the regularity and transverse orientation requirements for the foliation, unstated. These conventions must be recovered before this becomes a fully specified mathematical statement.

Short English statement

Nariman and Yazdi ask whether a pair of oppositely signed (co)oriented contact structures whose contact invariants pair to $1$ forces the underlying manifold to admit a Reebless foliation. The supplied record leaves essential manifold and invariant conventions unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800066$, exact source problem code catalogue entry, titled "Foliations and Diffeomorphism Groups --- Question 7." The registry attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open; this status is supplied metadata, not an independently verified assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 7. Supplied source URL: https://arxiv.org/abs/2503.18274. The provenance records access on 2026-10-16 and extraction method arxiv_latex. These are supplied provenance claims; the source was not independently consulted for this draft.