Legendrian non-isotopic unit conormal bundles in high dimensions

For any compact connected submanifold $K$ of $\mathbb{R}^n$, let $Λ_K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of $\mathbb{R}^n$. In this paper, we give examples of pairs $(K_0,K_1)$ of compact connected submanifolds of $\mathbb{R}^n$ such that $Λ_{K_0}$ is not Legendrian isotopic to $Λ_{K_1}$, although they cannot be distinguished by classical invariants. Here, $K_1$ is the image of an embedding $ι_f \colon K_0 \to \mathbb{R}^n$ which is regular homotopic to the inclusion map of $K_0$ and the codimension in $\mathbb{R}^n$ is greater than or equal to $4$. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for $Λ_K$ when the codimension of $K$ is greater than or equal to $4$. The main examples $Λ_{K_0}$ and $Λ_{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.

💡 Research Summary

The paper investigates the Legendrian isotopy classification of unit conormal bundles Λ_K associated to compact connected submanifolds K⊂ℝⁿ when the codimension d = n‑dim K is at least four. Classical Legendrian invariants—the Thurston‑Bennequin number, the Maslov class, and the smooth isotopy class of K—are shown to be insufficient for distinguishing Λ_{K₀} from Λ_{K₁} in this high‑dimensional setting. To overcome this limitation the author introduces a new non‑classical invariant: strip Legendrian contact homology (SLCH) together with a coproduct operation defined on its homology.

The construction begins with the contact manifold (U⁎ℝⁿ,α) where U⁎ℝⁿ is the unit cotangent bundle equipped with its canonical contact form. For a Legendrian submanifold Λ⊂U⁎ℝⁿ, a suitable almost complex structure J on ℝ×U⁎ℝⁿ is chosen satisfying a positivity condition (denoted (⋆)). The chain complex (C L⁎(Λ), d_J) is generated by Reeb chords of Λ; the differential d_J counts J‑holomorphic strips with one positive and one negative puncture. This yields the strip Legendrian contact homology LCH⁎(Λ). The coproduct δ_J is defined by counting J‑holomorphic curves with one positive and two negative punctures; the (⋆) condition guarantees transversality and compactness of the relevant moduli spaces.

When Λ = Λ_K for a submanifold K of codimension d≥4, the author proves that SLCH is canonically isomorphic to the relative homology H_{+1‑d}(K×K,Δ_K;ℤ/2), where Δ_K is the diagonal. This isomorphism is constructed via Morse theory on K×K and via a detailed analysis of pseudo‑holomorphic disks with switching Lagrangian boundary conditions (the conormal bundle and the zero section). Moreover, under this identification the coproduct δ_J corresponds exactly to a topologically defined coproduct δ_K on H_{+1‑d}(K×K,Δ_K;ℤ/2). The map δ_K is described both in singular homology (using a Thom‑class construction) and in Morse homology (using gradient flow trees), and it is shown to be the homology‑level incarnation of a string‑topology operation on the space of paths with endpoints on K.

The central topological theorem (Theorem 1.3) states: if Λ_K and Λ_{K’} are Legendrian isotopic and both have codimension ≥4, then there exists a ℤ/2‑linear isomorphism Θ:H_(K)→H_(K’) intertwining the coproducts, i.e. (Θ⊗Θ)∘δ_K = δ_{K’}∘Θ. Consequently, any difference in the coproduct structure provides an obstruction to Legendrian isotopy.

To produce explicit examples, the author fixes integers n and d with d≥4 and n≥2d, chooses a compact connected manifold M⊂ℝ^{n‑d}, and sets K₀ = S^{d‑1}×M ⊂ ℝⁿ. An embedding ι_f : K₀ → ℝⁿ is constructed as a connected sum of K₀ with a standard sphere S^{n‑d} embedded in ℝⁿ; the image is denoted K₁. Both K₀ and K₁ have the same classical invariants, and for d≥3 their complements are simply connected, so previous algebraic invariants (e.g., π₁ of the complement) cannot distinguish them. However, the coproducts behave differently: δ_{K₀}=0 for any M, while δ_{K₁}≠0 whenever H_k(M;ℤ/2)=0 for some 1≤k≤n‑2d. By Theorem 1.3 this forces Λ_{K₀} and Λ_{K₁} to be non‑Legendrian‑isotopic (Theorem 1.1). Moreover, Theorem 1.2 computes the dimensions of LCH in degree 2k+2d‑4 and shows a strict inequality between the two examples, providing a quantitative distinction.

The paper also discusses potential extensions: lifting δ_K to an integral (ℤ) map δ_ℤ using the Thom class of the oriented normal bundle, and incorporating spin structures to define SLCH over ℤ. This would align the construction with the Chekanov–Eliashberg DGA over ℤ