Thirty Years of Floer Theory for 3-manifolds
Thirty Years of Floer Theory for 3-manifolds
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三流形 Floer 理论三十年
DOI：
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Robert Lipshitz
中科院分区：
文献类型：
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作者：
Jennifer Hom;Robert Lipshitz
Floer同调，由Andreas Floer在1988年提出，对低维拓扑学领域产生了巨大的影响，特别是通过它的化身，如Heegaard Floer同调。这次研讨会的目的是汇集了该领域的领导者和正在崛起的年轻研究人员，讨论Floer同源性的应用和关系，手术问题，基本组和量子不变量。Heegaard Floer同调（英语：Heegaard Floer homology），由Ozsváth-Szabó在21世纪初定义，是三维和四维流形的一组不变量，也是其中嵌入的纽结和曲面的不变量。在其最简单的形式中，Heegaard Floer同调将一个分次向量空间<$F（Y）关联到一个闭三维流形Y。Y中的零同调纽结K诱导Heegaard Floer复形上的一个滤子，而相应的分次复形的同调是K的纽结Floer同调，它是亚历山大多项式的范畴.虽然亚历山大多项式提供了结属的界限和对重叠的阻碍，但结Floer同源性实际上检测结属和重叠。纽结-弗洛尔复合体的一个特别好的特征是，它实际上决定了所有理性手术沿着Y中的K的Heegaard-弗洛尔同调。Heegaard Floer同调和纽结Floer同调是理解同调配边和纽结协调的有力工具，因为它们产生了大量的同调配边和纽结协调的代数不变量。一个统一的观点，从其中看到这些不变量都是定义良好的来自最近的工作Zemke，谁表明，某些装饰协边映射诱导特别好的地图上相关的Heegaard Floer不变量。这个观点也可以推广到亨德里克斯-马诺列斯库的对合弗洛尔理论。一个有趣的问题是考虑Heegaard Floer同调如何与更经典的不变量（如基本群）相关。Boyer-Gordon-沃森指出，对于一个封闭的、不可约的、有理同调球面Y，以下是等价的：
Floer homology, introduced by Andreas Floer in 1988, has had a dramatic impact on the field of lowdimensional topology, particularly via its incarnations as monopole and Heegaard Floer homologies. The purpose of this workshop was to bring together both leaders in the field and rising young researchers to discuss applications and relations of Floer homology to surgery problems, the fundamental group, and quantum invariants. Heegaard Floer homology, defined by Ozsváth-Szabó in the early 2000s, is a package of invariants for 3and 4-dimensional manifolds, as well as for embedded knots and surfaces inside them. In its simplest form, to a closed 3-manifold Y , Heegaard Floer homology associates a graded vector-space ĤF (Y ). A null-homologous knot K in Y induces a filtration on the Heegaard Floer complex, and the homology of the associated graded complex is the knot Floer homology of K, which categorifies the Alexander polynomial. While the Alexander polynomial provides bounds on the knot genus and obstructions to fibering, knot Floer homology in fact detects both the knot genus and fibering. One particularly nice feature of the knot Floer complex is that it actually determines the Heegaard Floer homology of all rational surgeries along K in Y . Heegaard Floer homology and knot Floer homology are powerful tools for understanding homology cobordism and knot concordance, as they yield a host of algebraic invariants of homology cobordism and knot concordance. A unified viewpoint from which to see that these invariants are all well-defined comes from recent work of Zemke, who shows that certain decorated cobordism maps induce particularly nice maps on the associated Heegaard Floer invariants. This viewpoint can also be promoted to the involutive Floer theory of Hendricks-Manolescu. An interesting question is to consider how Heegaard Floer homology is related to more classical invariants such as the fundamental group. Boyer-Gordon-Watson conjectured that for a closed, irreducible, rational homology sphere Y , the following are equivalent:
Florer 简单流形和 L 空间区间
影响因子：
1.7
作者：
Rasmussen J
通讯作者：
Rasmussen J
影响因子：
2.5
作者：
Hedden, Matthew;Kim, Se-Goo;Livingston, Charles
通讯作者：
Livingston, Charles
影响因子：
1.5
作者：
Hedden, Matthew;Levine, Adam Simon
通讯作者：
Levine, Adam Simon