Algebraic and Geometric Overtwistedness

Definitions and mathematical statement

A contact structure on a smooth manifold $M$ of dimension $2n+1$ is a hyperplane distribution $\xi\subset TM$ locally expressible as $\ker\alpha$, where $\alpha\wedge(\mathrm d\alpha)^n$ is nowhere zero. This local definition does not impose coorientability. The supplied record does not specify the allowed dimensions, compactness, boundary conditions, or coorientability assumptions.

The supplied question asks whether algebraic overtwistedness and overtwistedness are equivalent for contact manifolds, emphasizing the implication from algebraic overtwistedness to overtwistedness. However, neither property is defined in the supplied record. In particular, no algebraic invariant, coefficient system, or criterion for algebraic overtwistedness is specified, and the intended geometric definition of overtwistedness is not stated.

To display the logical target without selecting missing definitions, let $\mathcal C$ denote the intended, unspecified class of contact manifolds, and let $A(M,\xi)$ and $O(M,\xi)$ be placeholders for the intended algebraic and geometric overtwistedness properties. The question has the formal shape \[ \forall (M,\xi)\in\mathcal C,\qquad A(M,\xi)\Longleftrightarrow O(M,\xi), \] with particular emphasis on \[ \forall (M,\xi)\in\mathcal C,\qquad A(M,\xi)\Longrightarrow O(M,\xi). \] These are formulation templates, not fully specified conjectures: $\mathcal C$, $A$, and $O$ require clarification from the source. The supplied literature assessment instead describes a truncated conditional conjecture about contact surgery, but no surgery construction, condition, or conclusion appears in the supplied statement. That discrepancy remains unresolved.

Short English statement

The supplied question asks whether algebraic overtwistedness is equivalent to geometric overtwistedness, especially whether the algebraic property forces the geometric one. The intended definitions and class of contact manifolds are missing, and the supplied contact-surgery assessment conflicts with the displayed question. The record labels the problem open; this draft does not independently verify that label.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001828$, title "Algebraic and Geometric Overtwistedness," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2011; no proposer is identified.
• Supplied source citation: Dynamische Systeme (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/34, pp. 1913--1970. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/34. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• The registry marks the statement as "exact" and supplies the verification claim that the official report confirms the statement as written. This is a supplied verification claim, not a source visit or verification performed for this draft.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem "open" and describes the record as a truncated conditional conjecture about contact surgery with insufficient context to identify a theorem; it asserts no resolution. Its listed checked source is the original Oberwolfach report linked above. The supplied background calls this assessment dated literature triage and requests independent verification of time-sensitive status and cited claims.