Lagrangian Embeddings of Closed Three-Manifolds

Definitions and mathematical statement

Identify ${\mathbb{C}}^3$ with ${\mathbb{R}}^6$ using coordinates $z_j=x_j+\sqrt{-1}y_j$ and its standard symplectic form \[ \omega_0=\sum_{j=1}^3{\,\mathrm{d}} x_j\wedge{\,\mathrm{d}} y_j. \] For a closed smooth real three-manifold $M$, a Lagrangian embedding is a smooth embedding $F:M\hookrightarrow{\mathbb{C}}^3$ such that $F^*\omega_0=0$. Closed means compact without boundary. The literal verified question asks \[ \forall\text{ closed }3\text{-manifolds }M\ \exists F:M\hookrightarrow{\mathbb{C}}^3, \qquad F^*\omega_0=0. \] No exactness, monotonicity or orientation condition is explicitly imposed. An immersion with self-intersections does not meet the embedding target.

The universal wording needs reconciliation with the exact-Lagrangian obstruction. Let $\lambda_0=\tfrac12\sum_j(x_j{\,\mathrm{d}} y_j-y_j{\,\mathrm{d}} x_j)$, so ${\,\mathrm{d}}\lambda_0=\omega_0$. Any Lagrangian embedding makes $F^*\lambda_0$ closed; if $H^1(M;{\mathbb{R}})=0$, this form is exact. The standard nonexistence of closed exact Lagrangians in standard ${\mathbb{C}}^3$ therefore obstructs the literal universal assertion, for example for $S^3$. No exactness hypothesis has been added to the source question: it follows in these cases from the cohomology. The intended unresolved classification must consequently be distinguished from the unqualified "every" formulation.

The literature assessment checked on 21 August 2026 reports embeddings after connected summing with $S^1\times S^2$ and restrictions for prime manifolds, without a general classification. Such connected-sum results change the manifold and do not imply an embedding of the original $M$; immersion constructions likewise do not answer the stated embedding question. The supplied open label and broad classification assessment cannot be treated as a verified claim that the literal universal assertion survives the obstruction above.

Short English statement

The literal question asks for a Lagrangian embedding of every closed three-manifold into standard ${\mathbb{C}}^3$. Its universal wording needs correction or qualification: manifolds with vanishing real first cohomology would force a forbidden closed exact Lagrangian. Connected-sum and immersion results concern different targets.

Sources

• Analysis and Topology in Interaction (2006), Oberwolfach Reports, DOI 10.4171/owr/2006/13, pp. 729--804, https://doi.org/10.4171/owr/2006/13.
• Literature assessment checked 21 August 2026: original report, https://doi.org/10.4171/owr/2006/13; "Ekholm Eliashberg Murphy Smith Lagrangian immersions," https://arxiv.org/abs/1303.0588; "connected sums of hyperbolic 3-manifolds," https://intlpress.com/site/pub/files/_fulltext/journals/mrl/2014/0021/0006/MRL-2014-0021-0006-a002.pdf.
• Local UnsolvedMath registry, permanent ID $30000442$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Corrected-and-verified status records notation normalization; the exact universal wording is retained for source review. The record supplies no separate full citation for the standard exact-Lagrangian nonexistence obstruction.