Unknotting Lagrangian $\mathrm{S}^1\times\mathrm{S}^{n-1}$ in $\mathbb{R}^{2n}$

TL;DR

The paper achieves a complete smooth isotopy classification of Lagrangian embeddings $\Sigma_n=S^1\times S^{n-1}$ into $\mathbb{C}^n$ (i.e. $\mathbb{R}^{2n}$ with the standard symplectic form) for $n\ge3$, revealing distinct behaviors by parity and dimension. The authors integrate a generalized Luttinger surgery framework to enforce linking constraints with push-offs, and a Haefliger–Hirsch $h$-principle analysis to reduce isotopy questions to homotopy data of frame maps, normal fields, and linking classes. They prove that for $n\neq2,4$, Lagrangian embeddings are smoothly isotopic to a standard model (Chekanov’s embedding) when $n$ is even, and to the model or its reflection when $n$ is odd, with precise exceptional cases explained: $n=2$ (totally different behavior in dimension four) and $n=4$ (two smooth isotopy classes both realized by Lagrangian embeddings). The results showcase a strong symplectic rigidity phenomenon: while totally real or soft embeddings admit many isotopy types, Lagrangian constraints drastically restrict the smooth categories, and the classification tightens further as $n$ grows, aligning with known rigidity phenomena in low dimensions and extending them to the general $n\ge3$ setting.

Abstract

Lagrangian embeddings $\mathrm{S}^1\times\mathrm{S}^{n-1}\hookrightarrow\mathbb{R}^{2n}$ are classified up to smooth isotopy for all $n\ge 3$.

Theorems & Definitions (22)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Lemma 2.5
• proof