Arnold's nearby Lagrangian conjecture

Theorem 1.1: for some sufficiently large even $N$ , $Q=S^9\times S^{N-1}$ admits a closed exact smoothly embedded Lagrangian $L\subset T^*Q$ , diffeomorphic to $Q$ and with $Q$ and $L$ simply connected, that no compactly supported Hamiltonian isotopy carries to the zero section. $L$ is cut out by a generating family whose data at infinity form a nontrivial class $a\in\pi_9$ of the stable smooth tube space with trivial associated spherical fibration; a Morse and h-cobordism argument shows such data cannot come from a family with one nondegenerate critical point per fibre, as an isotopy to the zero section would force. Not shown: an explicit $N$ , counterexamples in low dimension, or any failure of the homotopy-theoretic consequences already proved.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Arnold's nearby Lagrangian conjecture · Nearby Lagrangian conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: Arnold's nearby Lagrangian conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract and Theorem 1.1 of the TeX source against the conjecture as the manuscript states it (every closed exact embedded Lagrangian in T*Q is Hamiltonian isotopic to the zero section). The theorem claims the negation for one base, so it addresses the conjecture as posed for all closed Q. No formalization exists for this family and the argument was not checked. Scope the paper itself states: N is some sufficiently large even integer, not given explicitly, so the counterexample lives in very high dimension; it says nothing about low-dimensional bases or specific bases such as spheres or tori. The proof leans on Waldhausen's tube fibration and Rognes's computation of low-degree smooth Whitehead groups to produce a nonzero class in the ninth homotopy group of the stable tube space with trivial image in BG. Given that the field broadly expected the conjecture to hold, an expert reading is especially wanted.

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Arnold's nearby Lagrangian conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (Invent. Math. 2012)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture (Geom. Topol. 2013)

paper · pending

paper link supplied by the VibeMathed registry.

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Arnold's nearby Lagrangian conjecture evidence for this event

VibeMathed record: Arnold's nearby Lagrangian conjecture evidence for this event

Arnold's nearby Lagrangian conjecture parent of this event

Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture (Geom. Topol. 2013) evidence for this event

Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (Invent. Math. 2012) evidence for this event

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