A counterexample to the nearby Lagrangian conjecture

1 · The essentials

The manuscript claims a counterexample to a general expectation about how certain geometric shapes can be moved into a standard position. Its example lives in a space built from a product of two spheres, one of dimension nine. The other sphere has a dimension determined by a sufficiently large even number. The constructed shape has the same smooth form as the underlying product of spheres. Nevertheless, the manuscript claims that it cannot be moved to the standard copy of that product using the particular kind of motion required by the conjecture. This challenges the unrestricted conjecture, not necessarily every version with additional conditions. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two arrangements of the same flexible object, with a rulebook restricting how you may move it. Having the same shape does not by itself guarantee that the permitted moves connect the arrangements; the manuscript claims a geometric example of this distinction.

Why it matters

The claimed example separates two questions: whether an object has the same smooth form as a standard one, and whether a prescribed geometric motion connects them. According to the supplied abstract, this distinction supplies a counterexample to the unrestricted nearby Lagrangian conjecture. The stated conclusion concerns that unrestricted claim rather than a classification of all possible examples.

2 · Going further

The precise claim concerns the cotangent bundle $T^{*} (S^{9} \times S^{N - 1})$ , for some sufficiently large even integer $N$ . The manuscript claims to construct a closed, exact, smoothly embedded Lagrangian $L$ in this space such that $L$ is diffeomorphic to $S^{9} \times S^{N - 1}$ but is not Hamiltonian isotopic to the zero section. The title and abstract identify this as a counterexample to the unrestricted nearby Lagrangian conjecture.

The decisive distinction in the statement is between being diffeomorphic and being Hamiltonian isotopic to the zero section. The former records sameness of smooth structure; the latter asks for a connection by a more restricted kind of geometric motion within the cotangent bundle. The claimed example retains the former property while lacking the latter.

The abstract states the construction and its claimed consequence, but gives no method of construction or argument for the failure of Hamiltonian isotopy. Consequently, the supplied material does not support an explanation of the argument's mechanism. Only the main manuscript is supplied, so there is no companion-manuscript relationship to describe.

Useful notions

Spheres and products of spaces
Smooth manifolds and diffeomorphisms
Cotangent bundles and the zero section
Exact Lagrangian embeddings
Hamiltonian isotopy

What was known before / what is new

Before

The source names the unrestricted nearby Lagrangian conjecture as the claim being challenged. It provides no historical account, prior partial results, or discussion of versions with additional hypotheses.

New

The manuscript claims a closed exact smoothly embedded Lagrangian in $T^{*} (S^{9} \times S^{N - 1})$ for some sufficiently large even $N$ . Its stated distinguishing feature is that it is diffeomorphic to the base but not Hamiltonian isotopic to the zero section. The abstract presents this as a counterexample to the unrestricted conjecture.

Limits and assumptions

The stated example requires some sufficiently large even integer $N$ ; the supplied abstract gives neither a value nor a threshold, and does not assert the construction for every even $N$ . The Lagrangian is required to be closed, exact, and smoothly embedded, and the ambient space is the specified cotangent bundle. No construction details or argument are supplied in the abstract. The main result is not formalized in Lean and has not been formally verified. The source does not specify which restricted versions of the conjecture remain open.

Mini glossary

Diffeomorphic: Related by a smooth invertible map whose inverse is also smooth.
Cotangent bundle: The space collecting the linear functionals on tangent vectors at every point of a smooth manifold.
Zero section: The standard copy of the base inside its cotangent bundle that assigns the zero linear functional to every point.
Closed exact embedded Lagrangian: A compact submanifold without boundary, embedded without self-intersections, of half the ambient dimension, on which the symplectic form vanishes and the canonical one-form is the differential of a function.
Hamiltonian isotopy: A smooth family of motions generated by Hamiltonian functions in the ambient symplectic space.

Keywords : Nearby Lagrangian conjecture · Exact Lagrangian · Hamiltonian isotopy

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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A counterexample to the nearby Lagrangian conjecture

23/09/2026

To be verified

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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