Lectures on Heegaard Floer Homology
Lectures on Heegaard Floer Homology
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Heegaard Floer 同调性讲座
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发表时间：
2005
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通讯作者：
P. Ozsváth
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作者：
P. Ozsváth
这些是2004年6月在克莱数学研究所布达佩斯暑期学校讲授的关于Heegaard Floer同调的第二堂课的笔记，由第一作者讲授。虽然该课程中涉及的一些主题没有纳入这些笔记（具体来说，在第一门课程的课堂笔记中描述的“结弗洛尔同源性”的讨论，参见。[44])的中心目标在很大程度上仍然是相同的：我们试图给一个相当直接的路径到一些拓扑应用的外科手术长的确切序列在Heegaard Floer同源。具体来说，目标是用最少的必要机械来证明Dehn手术对解开结的描述，这是与Peter Kronheimer，Tomasz Mrowka和Zoltán Szabó合作首次建立的。(This这个问题在[29]中首次使用SeibergWitten规范理论解决，而不是Heegaard Floer同调;这里概述的方法可以在[39]中找到。在第一讲中，阐述了手术精确三角形，并给出了它的一些直接应用。在第二讲中，这一点得到了证明。第三讲讨论由三流形之间的光滑协边诱导的映射。这是包含最少技术细节的讲座-尽管大部分可以在[34]中找到。在第四课中，我们展示了精确的三角形，以及其中出现的映射的性质，如何证明了Dehn手术分类的解结。已试图使讨论尽可能简单。例如，在这些注释中，我们避免使用"扭曲系数"。这是有代价的：因此，我们没有开发出必要的机械来处理第一种类型的结。我们希望读者的兴趣将被充分激发，以研究原始文件，以填补这一空白。还有一些练习分散在整个文本中，主题从同调代数和初等保形映射到低维拓扑。强烈鼓励读者通过这些练习进行思考;文中的一些证明依赖于它们。在每堂课的结尾，有一个关于材料进一步阅读的讨论。
These are notes for the second lecture course on Heegaard Floer homology in the Clay Mathematics Institute Budapest Summer School in June 2004, taught by the first author. Although some of the topics covered in that course did not make it into these notes (specifically, the discussion of “knot Floer homology” which instead is described in the lecture notes for the first course, cf. [44]), the central aim has remained largely the same: we have attempted to give a fairly direct path towards some topological applications of the surgery long exact sequence in Heegaard Floer homology. Specifically, the goal was to sketch with the minimum amount of machinery necessary a proof of the Dehn surgery characterization of the unknot, first established in a collaboration with Peter Kronheimer, Tomasz Mrowka, and Zoltán Szabó. (This problem was first solved in [29] using SeibergWitten gauge theory, rather than Heegaard Floer homology; the approach outlined here can be found in [39].) In Lecture 1, the surgery exact triangle is stated, and some of its immediate applications are given. In Lecture 2, it is proved. Lecture 3 concerns the maps induced by smooth cobordisms between three-manifolds. This is the lecture containing the fewest technical details – though most of those can be found in [34]. In Lecture 4, we show how the exact triangle, together with properties of the maps appearing in it, lead to a proof of the Dehn surgery classification of the unknot. An attempt has been made to keep the discussion as simple as possible. For example, in these notes we avoid the use of “twisted coefficients”. This comes at a price: as a result, we do not develop the necessary machinery required to handle knots with genus one. It is hoped that the reader’s interest will be sufficiently piqued to study the original papers to fill in this gap. There are also a number of exercises scattered throughout the text, in topics ranging from homological algebra and elementary conformal mapping to low-dimensional topology. The reader is strongly encouraged to think through these exercises; some of the proofs in the text rely on them. At the conclusion of each lecture, there is a discussion on further reading on the material.