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A counterexample to the nearby Lagrangian conjecture
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Abstract
We disprove the unrestricted nearby Lagrangian conjecture. For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.
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Comments and discussion
- Wenyuan Li, Nearby Lagrangian conjecture, present and future (ZUMA-GT seminar abstract, 2026-10-15). Announces a discussion of OpenAI's proposed embedded counterexample through generating families and parametrized h-cobordism spaces, followed by remaining questions and a perspective from C⁰-limits explicitly conditional on the counterexample being correct.
Possibly relevant literature (found by the agent, not checked)
- Tobias Ekholm, Thomas Kragh, Ivan Smith, Lagrangian exotic spheres (arXiv:1503.00473, 2015). Shows that some reparametrizations of the zero section of $T^*S^n$, viewed as maps, are not Hamiltonian isotopic to the original zero section; this is an earlier non-isotopy phenomenon to compare with Theorem 1.1, which concerns the embedded submanifold $L$.
- Mohammed Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres (arXiv:0812.4781, 2008; accepted by Annals of Mathematics per arXiv comments). Constrains the smooth type of exact Lagrangians in $T^*S^n$; related to the paper's point that the obstruction lies in how $L$ sits in $T^*Q$, not in its abstract smooth structure, which agrees with $Q$.
- Nikolas Adaloglou, Gerard Bargalló i Gómez, Johannes Hauber, The nearby Lagrangian conjecture for pinwheels (arXiv:2605.22473, May 2026). Proves a Hamiltonian-isotopy uniqueness statement of nearby-Lagrangian type for singular Lagrangian skeleta in rational homology balls; this is a recent positive case to read alongside the low-dimensional results surveyed in the introduction.
Digests and expositions
- Latent Seminar, A counterexample to the nearby Lagrangian conjecture (expository article, 2026-10-07). Explains the manuscript's stable tube class, momentum-separation argument, one-critical-point obstruction and Hamiltonian transport, while explicitly stating that it does not independently verify the complete proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 340
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{A counterexample to the nearby Lagrangian conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026}},
year = {2026}
}Family 340: A counterexample to the nearby Lagrangian conjecture
Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.
This is the only paper in this family.