Topological Invariance of Helicity

Definitions and mathematical statement

The supplied record asks whether helicity is preserved by topological conjugacy of exact volume-preserving vector fields. It does not specify the ambient manifold, its boundary conditions, the volume form, or whether the conjugating homeomorphism must preserve volume. Consequently, the intended quantified statement cannot be recovered uniquely.

For a conditional formulation in the standard closed three-dimensional setting, let $M$ be a smooth closed oriented three-manifold with a smooth positive volume form $\mu$. A smooth vector field $X$ preserves volume if $\mathcal L_X\mu=0$, and is exact volume-preserving if there exists a smooth one-form $\alpha$ such that \[ \iota_X\mu={\,\mathrm{d}}\alpha. \] Its helicity relative to $\mu$ is \[ \mathcal H_\mu(X)=\int_M\alpha\wedge{\,\mathrm{d}}\alpha. \] In this closed setting the integral is independent of the choice of primitive $\alpha$. Write $\varphi_X^t$ for the flow of $X$. A time-preserving topological conjugacy between $X_1$ and $X_2$ is a homeomorphism $h:M\to M$ satisfying \[ h\circ\varphi_{X_1}^t=\varphi_{X_2}^t\circ h \qquad\text{for every }t\in{\mathbb{R}}. \] Under this conditional interpretation, the question is whether, for every such $(M,\mu)$ and every pair of smooth exact volume-preserving vector fields $X_1,X_2$ on $M$, the existence of a topological conjugacy implies \[ \mathcal H_\mu(X_1)=\mathcal H_\mu(X_2). \] The source does not establish that this choice of manifold, regularity, common volume form, or unrestricted conjugating homeomorphism is its intended scope. In particular, whether one must additionally require $h_*\mu=\mu$, or impose orientation conditions, remains unspecified. Manifolds with boundary would require additional hypotheses before the displayed helicity definition could be used without qualification.

The supplied literature assessment dated 2026-08-22 labels the problem partially solved: it reports invariance and a topological extension for nowhere-vanishing exact volume-preserving flows, while describing the full question allowing fixed points as unresolved. These are supplied curation claims, not independently verified conclusions. The full source question therefore retains the possibility $X_i(p)=0$; no nowhere-vanishing assumption is imposed on it.

Short English statement

The question asks whether exact volume-preserving flows related by a time-preserving homeomorphism must have equal helicity, including when fixed points occur. The source leaves the ambient geometry and requirements on the conjugacy unspecified, so the displayed closed-three-manifold formulation is conditional. Supplied dated curation reports a result for nowhere-vanishing flows and leaves the general case unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006363$, source problem code catalogue entry, title Topological Invariance of Helicity; registry collection data/unsolvedmath/problems.json, with additional existing records in catalog/problems.json.
• Supplied report citation: Dynamische Systeme (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/31, pp. 1647--1706. https://doi.org/10.4171/owr/2025/31. The original statement attributes the presentation to Oliver Edtmair, joint work with Sobhan Seyfaddini; the separate proposer field is empty.
• Edtmair--Seyfaddini, Universal extension of helicity. Supplied literature link: https://arxiv.org/abs/2508.10609.
• Oberwolfach Report 31/2025. Supplied literature link: https://ems.press/content/serial-article-files/52240.
• Literature assessment checked at 2026-08-22: supplied dated curation labels the problem partially solved, reports conjugacy invariance and a topological extension for nowhere-vanishing exact volume-preserving flows, and describes the case allowing fixed points as unresolved. This assessment and its cited claims have not been independently verified here.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. This is supplied source-verification metadata, not a claim of verification in this draft.