A Duality Exact Sequence for Legendrian Contact Homology
A Duality Exact Sequence for Legendrian Contact Homology
复制标题
Legendrian接触同调的对偶精确序列
DOI：
10.1215/00127094-2009-046
复制
发表时间：
2008
期刊：
影响因子：
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通讯作者：
J. Sabloff
中科院分区：
文献类型：
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作者：
T. Ekholm;John B. Etnyre;J. Sabloff
我们建立了P x R中的Legendrian子流形L的一个长精确序列，其中P是一个精确辛流形，它允许哈密顿同位素取代L自身的投影。在这个序列中，奇异同调H_*映射到线性化的接触上同调CH_*， CH_*映射到线性化的接触同调CH_*， CH_*映射到奇异同调。特别地，这个序列意味着映射的核（CH_*\到H_*）和映射的核（H_* \到CH^*）之间的对偶性。更进一步，这个对偶在以下意义上与L中的庞加莱对偶相容：一个奇异类的庞加莱对偶是CH_*中a的像映射到CH^*中的类\ α，使得\ α (a)=1。
We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \alpha in CH^* such that \alpha(a)=1.
The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].