The plastikstufe – a generalization of the overtwisted disk to higher dimensions
The plastikstufe – a generalization of the overtwisted disk to higher dimensions
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plastikstufe——过度扭曲圆盘向更高维度的推广
DOI：
10.2140/agt.2006.6.2473
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发表时间：
2006
影响因子：
0.7
通讯作者：
Klaus Niederkrüger
中科院分区：
文献类型：
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作者：
Klaus Niederkrüger
接触拓扑的情况可以粗略地表述为：通过拓扑方法可以非常充分地理解3维接触流形，已经实现了深远的分类，并且已经建立了与许多其他领域的关系。相比之下，高维接触几何的世界地图几乎完全由白点组成。构建此类流形的有力方法是接触手术，迄今为止开发的区分不同接触结构最有前途的技术是接触同源性，并且通过Giroux的开卷分解，希望能够获得一些分类结果。接触 3 流形的第一个结构区别是过度扭曲的概念。事实证明，这样的流形首先不允许 Eliashberg [4] 和 Gromov [8] 进行（甚至是弱的）辛填充，其次可以像 Eliashberg [3] 那样以非常令人满意的方式进行分类。令人惊讶的是，在更高维度中，尚未发现类似的标准。吉鲁根据他的开书分解提出了一个定义，该定义在三个维度上与标准维度完全等效。相比之下，我们的定义是基于塑料的存在，是过度扭曲圆盘的直接概括。在格罗莫夫关于全纯曲线的著名论文 [8] 中，给出了对某物（可能是塑料）的粗略描述。本文中描述的过度扭曲的概括是由 Yuri Chekanov 独立发现的。有趣的是，他的（未发表的）定理 1 证明使用了非常不同的方法。
The situation of contact topology can be roughly stated like this: the 3‐dimensional contact manifolds can be understood very adequately by topological methods, a farreaching classification has been achieved and relations to many other fields have been established. In contrast, the world map of higher-dimensional contact geometry consists almost entirely of white spots. A powerful method for constructing such manifolds is contact surgery, the most promising technique developed so far to distinguish different contact structures is contact homology and with Giroux’s open book decomposition, it is hoped that some classification results could be obtained. The first structural distinction found for contact 3‐manifolds was the notion of overtwistedness. It turned out that such manifolds firstly do not allow an (even weak) symplectic filling by Eliashberg [4] and Gromov [8], and secondly can be classified in a very satisfactory way as in Eliashberg [3]. In higher dimensions, surprisingly, no analogous criterion has yet been found. Giroux has proposed a definition based on his open book decomposition, which in three dimensions is completely equivalent to the standard one. In contrast, our definition is based on the existence of a plastikstufe, a direct generalization of the overtwisted disk. In Gromov’s famous paper on holomorphic curves [8], a sketchy description of something, which possibly could be a plastikstufe, is given. The generalization of overtwistedness described in this article was found independently by Yuri Chekanov. Interestingly, his (unpublished) proof of Theorem 1 uses very different methods.