Let N and L be closed, simply connected manifolds of the same dimension. The cotangent bundle T^N carries its natural symplectic structure, and consider the space of Lagrangian embeddings of L into T^N. Homotopy-theoretic Nearby Lagrangian conjecture. The space of Lagrangian embeddings of L in T^N is contractible if L is diffeomorphic to N, and is empty otherwise. This is a weaker homotopy-theoretic version of Arnold's Nearby Lagrangian conjecture, which concerns Lagrangian submanifolds of cotangent bundles and remains open. The conjecture is motivated by recent homotopy-theoretic and Floer-theoretic results.

The homotopy-theoretic Nearby Lagrangian conjecture

The conjecture predicts that the space of Lagrangian embeddings is empty unless the source and base are diffeomorphic, and contractible when they are. It is formulated as Conjecture 1.1 in the catalogued work, which reports no resolution.

Known results

For a closed exact Lagrangian in $T^*N$ , projection to $N$ is a simple homotopy equivalence; related work traces the homotopy-equivalence result to Abouzaid.
The conjecture is known when the base is $S^2$ , via work of Hind, Hind–Pinton–Wendl, and others.
For Arnold’s stronger nearby Lagrangian conjecture, related results establish homotopy, fundamental-group, and stable-Gauss-map constraints, but not contractibility of the embedding space.

Current status (as of August 2026): The homotopy-theoretic conjecture remains open in general; no claimed proof, counterexample, or verification was found, although some individual-Lagrangian consequences and low-dimensional cases are established.

The homotopy-theoretic Nearby Lagrangian conjecture

Primary source

Progress summary

Solutions 0