Floer homology and the symplectic isotopy problem
Floer homology and the symplectic isotopy problem
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弗洛尔同源性和辛同位素问题
DOI：
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发表时间：
1997
期刊：
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通讯作者：
P. Seidel
中科院分区：
文献类型：
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作者：
P. Seidel
辛合痕问题是关于紧致辛流形的自同构的问题。它问的关系，辛合痕之间的这种自同构是精细的关系，比光滑合痕（光滑）。这篇论文的主要结果是，有辛流形的答案是积极的，事实上，一大类辛四流形被证明具有这一性质。这个结果是研究一类特殊的辛自同构的结果，称为广义Dehn扭曲。研究辛合痕问题的难点在于如何证明两个给定的辛自同构不是辛合痕的。辛弗洛尔同调理论赋予任何辛自同构一个“同调群”。这些群在辛合痕下是不变的，因此是该任务的明显候选者。虽然没有计算弗洛尔同调群的一般程序，但事实证明，这对广义德恩扭曲是可行的。计算涉及的函结构的Floer同调群的扩展：我们引入同态引起的某些辛纤维奇异性。然后，我们使用的事实，广义德恩扭曲出现单值映射的纤维化。这些诱导映射的Floer同调群可能是感兴趣的独立的贡献辛合痕问题。论文分为三个部分：第一部分介绍了辛合痕问题，介绍了广义Dehn扭，并解释了确定其Floer同调群的后果。第二部分致力于Floer同调，其主要重点是新的诱导映射。最后一部分描述了广义Dehn扭的Floer同调群的计算。
The symplectic isotopy problem is a question about automorphisms of a compact symplectic manifold. It asks whether the relation of symplectic isotopy between such automorphism is finer than the relation of diffeotopy (smooth isotopy). The principal result of this thesis is that there are symplectic manifolds for which the answer is positive; in fact, a large class of symplectic four-manifolds is shown to have this property. This result is the consequence of the study of a special class of symplectic automorphisms, called generalized Dehn twists. The hard part of studying the symplectic isotopy problem is how to prove that two given symplectic automorphisms are not symplectically isotopic. Symplectic Floer homology theory assigns a ‘homology group’ to any symplectic automorphism. These groups are invariant under symplectic isotopy, hence an obvious candidate for the task. While there is no general procedure for computing the Floer homology groups, it turns out that this is feasible for generalized Dehn twists. The computation involves an extension of the functorial structure of the Floer homology groups: we introduce homomorphisms induced by certain symplectic fibrations with singularities. Then we use the fact that generalized Dehn twists appear as monodromy maps of such fibrations. These induced maps on Floer homology groups may be of interest independently of their contribution to the symplectic isotopy problem. The thesis is divided into three parts: the first part presents the symplectic isotopy problem, introduces generalized Dehn twists, and explains the consequences of the determination of their Floer homology groups. The second part is devoted to Floer homology; its main focus are the new induced maps. The final part describes the computation of the Floer homology groups of a generalized Dehn twist.