Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications
Lifting pseudo-holomorphic polygons to the symplectisation of $P \times \mathbb{R}$ and applications
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DOI：
10.4171/qt/73
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发表时间：
2013-05
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通讯作者：
Georgios Dimitroglou Rizell
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作者：
Georgios Dimitroglou Rizell
设$\mathbb{R} \乘以（P \乘以\mathbb{R}）$是精确辛流形$P$的接触的辛化，并且设$\mathbb{R} \乘以\Lambda$是接触中的一个勒让子流形上的柱体。我们证明了$P$中具有$\Lambda$投影上边界的伪全纯多边形可以提升为具有$\mathbb{R} \乘以\Lambda$上边界的辛化中的伪全纯圆盘。由此可见，Legendrian接触同调可以通过计算这两个对象中的任意一个来等价地定义。利用这一结果，我们证明了由精确拉格朗日填充引起的线性化Legendrian接触同构的Seidel同构以及该填充的奇异同构。
Let $\mathbb{R} \times (P \times \mathbb{R})$ be the symplectisation of the contactisation of an exact symplectic manifold $P$, and let $\mathbb{R} \times \Lambda$ be a cylinder over a Legendrian submanifold in the contactisation. We show that a pseudo-holomorphic polygon in $P$ having boundary on the projection of $\Lambda$ can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on $\mathbb{R} \times \Lambda$. It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel's isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling.