论文标题
倾向于单位综合束的Legendrian接触同源性的拓扑描述
Toward a topological description of Legendrian contact homology of unit conormal bundles
论文作者
论文摘要
对于Riemannian歧管$ Q $的平滑紧凑型submanifold $ k $，其单位套件$λ_k$是带有规范接触结构的单位cotangent bundle $ q $的legendrian submanifold。使用伪形构曲线技术，$λ_K$的Legendrian触点同源性在例如$ q = \ mathbb {r}^n $时。在本文中，旨在给出此同源性的另一个描述，我们为任何一对$（q，k）$的分级$ \ mathbb {r} $ - 代数从字符串拓扑的角度使用方向定义，并在$ k $的光滑同位素下证明其不变性。作者猜想是，在所有学位上，$λ_k$的legendrian触点同源物和系数在$ \ mathbb {r} $中是同构的。这是Cieliebak，Ekholm，Latschev和Ng引入的同源组的重新构造，当时$ K $的编辑为$ 2 $，尽管该系数从原始$ \ Mathbb {Z}降低，但该系数降低了。我们在特定示例中计算所有学位的不变（i），并且（ii）$ 0 $ tem $ k $的$ 2 $ 2 $ 2 $ - 平面捆绑包时（ii）。
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.
