arXiv 0903.07682009-03-04math.SG

Rabinowitz弗洛尔同调与辛同调

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea

备注：: 59 pages, 8 figures

摘要：

Rabinowitz-弗洛尔同调群$RFH_*(M,W)$与接触流形$(M,ξ)$到辛流形$(W,ω)$的精确嵌入相关联。它们仅依赖于$W\setminus M$的有界分量$V$。我们构造了一个长正合序列，其中$V$的辛上同调映射到$V$的辛同调，后者又映射到Rabinowitz-弗洛尔同调$RFH_*(M,W)$，后者又映射到$V$的辛上同调.我们计算了$RFH_*(ST^*L,T^*L)$，其中$ST^*L$是闭流形$L$的单位共球丛。作为一个应用，我们证明了一个精确的接触嵌入的$ST^*L$（赋予标准的接触结构）的图像不能被移位远离自身的哈密顿合痕，提供$\dim L\ge 4$和嵌入诱导注入$π_1$。特别地，如果$L$是单连通的，则$ST^*L$不允许精确接触嵌入到亚临界Stein流形中。我们还证明了Weinstein猜想在辛流形中成立，辛流形允许精确的可移位余维0嵌入。

英文摘要：

The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,ξ)$ into a symplectic manifold $(W,ω)$. They depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz-Floer homology $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, where $ST^*L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact embedding of $ST^*L$ (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided $\dim L\ge 4$ and the embedding induces an injection on $π_1$. In particular, $ST^*L$ does not admit an exact contact embedding into a subcritical Stein manifold if $L$ is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings.

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