Toward a topological description of Legendrian contact homology of unit conormal bundles

TL;DR

For a smooth oriented manifold $Q$ and a compact oriented submanifold $K\subset Q$, this work defines a unital graded $\mathbb{R}$-algebra $H^{\mathrm{string}}_*(Q,K)$ via de Rham chains on differentiable path spaces and an inverse (and direct) limit construction over length parameters. The authors show $H^{\mathrm{string}}_*(Q,K)$ is invariant under smooth isotopies of $K$ and provide a product structure induced by path concatenation, aiming to realize a topological description of Legendrian contact homology of the unit conormal bundle $\Lambda_K$ through a real-coefficient framework. In particular, they connect to the cord algebra in codimension two with trivial normal bundle, and prove explicit computations in Euclidean spaces: $H^{\mathrm{string}}_*(\mathbb{R}^{2d-1},K_0\cup K_1)$ matches a Hopf-type DGA and $H^{\mathrm{string}}_*(\mathbb{R}^{2d-1},K_0\cup K_2)$ matches an unlink-type algebra, illustrating how path-space topology encodes Legendrian invariants. A central conjecture proposes an isomorphism with Legendrian contact homology $HC^{\mathrm{Legendrian}}_*(UT^*\mathbb{R}^n,\Lambda_K)$ with coefficients in $\mathbb{R}$ for all degrees, which would yield a purely topological route to study Legendrian contact topology via $H^{\mathrm{string}}_*(Q,K)$.

Abstract

For a smooth compact submanifold $K$ of a Riemannian manifold $Q$, its unit conormal bundle $Λ_K$ is a Legendrian submanifold of the unit cotangent bundle of $Q$ with a canonical contact structure. Using pseudo-holomorphic curve techniques, the Legendrian contact homology of $Λ_K$ is defined when, for instance, $Q=\mathbb{R}^n$. In this paper, aiming at giving another description of this homology, we define a graded $\mathbb{R}$-algebra for any pair $(Q,K)$ with orientations from a perspective of string topology and prove its invariance under smooth isotopies of $K$. The author conjectures that it is isomorphic to the Legendrian contact homology of $Λ_K$ with coefficients in $\mathbb{R}$ in all degrees. This is a reformulation of a homology group, called string homology, introduced by Cieliebak, Ekholm, Latschev and Ng when the codimension of $K$ is $2$, though the coefficient is reduced from original $\mathbb{Z}[π_1(Λ_K)]$ to $\mathbb{R}$. We compute our invariant (i) in all degrees for specific examples, and (ii) in the $0$-th degree when the normal bundle of $K$ is a trivial $2$-plane bundle.

Figures (6)

• Figure 1: The process to define $\widetilde{\gamma}^1_k$ and $\widetilde{\gamma}^2_k$.
• Figure 2: The case where $(\gamma_l)_{l=1,\dots ,m}$ intersects both $(\sigma_i)_{i=1,2}$ and $(\sigma'_i)_{i=1,2}$.
• Figure 3: The red path is $\varphi(p,p'_{(w,p)} )$. The gray region is the tubular neighborhood $N_{\varepsilon}$ of $K_0\cup K_1$.
• Figure 4: The red path is $\varphi(p,p')$ when $p$ is close to $p_0$ or $p'$ is close to $p_1$. The gray region is the tubular neighborhood $N_{\varepsilon}$ of $K_0\cup K_1$
• Figure 5: The picture of $\widehat{\gamma}$ defined for $(\gamma)_{k=1}\in \Sigma^{\infty}_1$.

Theorems & Definitions (125)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Remark 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Example 2.6