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Lagrangian Floer cohomology and Khovanov homology

拉格朗日弗洛尔上同调和科万诺夫同调

基本信息

批准号：
EP/H035303/1
负责人：
Dominic Joyce
金额：
$ 47.63万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH035303%2F1
关键词：
Lagrangian Floer cohomology Khovanov homology

项目摘要

Most of modern geometry studies some kind of space. The spaces considered in differential geometry are called manifolds , spaces which locally look like n-dimensional Euclidean space but globally have an interesting shape. A manifold is compact if it is closed up, with no edges. The surface of a doughnut is a compact 2-dimensional manifold. A submanifold N of a manifold M is a subset of M which is itself a manifold, usually of smaller dimension than M. There are two kinds: embedded submanifolds, which may not intersect (cross) themselves, and immersed submanifolds, which may.One usually considers manifolds with some extra geometric structure, such as a Riemannian metric , which tells you the lengths of paths in the manifold, or a symplectic structure , which tells you the areas of 2-dimensional submanifolds. Symplectic manifolds are the foundation of the mathematical formulation of mechanics, and so of much of classical physics. They are also very interesting in their own right. Mathematicians like them as they are one of very few structures with an infinite-dimensional amount of symmetry, which gives symplectic geometry an unusual, entirely global flavour. Lagrangian submanifolds are a special kind of submanifold of a symplectic manifold. Given two compact, embedded Lagrangian submanifolds L, L* of a symplectic manifold M, one can under certain conditions define the Floer cohomology groups HF(L,L*), which are roughly speaking finite-dimensional vector spaces. The definition is very difficult. To do it, one chooses an auxiliary complex structure J on M and counts J-holomorphic 2-dimensional discs D in M with boundary (edge) in the union of L and L*. The remarkable thing about HF(L,L*) is that it is independent of the choice of J, and is also unchanged by moving L and L* around amongst Lagrangian submanifolds. It encodes some mysterious, nontrivial information about Lagrangian submanifolds one cannot get at in any other known way. It is a powerful tool in symplectic geometry. In previous EPSRC-funded research, the PI and Akaho extended the definition of HF(L,L*) from embedded to immersed Lagrangians. The PI also developed new technology ( Kuranishi (co)homology ) which will simplify and streamline the definition of HF(L,L*).This proposal will exploit these ideas. We will first develop a new, simpler and more general formulation of HF(L,L*), for immersed L,L*, using the PI's new technology. Then we will apply this new formulation to four problems. The first problem will prove a conjecture about HF(L,L*) when L,L* are complex Lagrangians in a hyperkahler manifold . The point is that the new version of HF(L,L*) will have technical features which make this proof much easier than with current definitions of HF(L,L*).The second and third problems concern knot theory: the study of knots (essentially, loops of string) in 3-dimensional space. Two knots K,K* are the same if you can deform K to K* without cutting the string. It is a difficult problem to compute whether two knots are the same. Mathematicians define knot invariants , numbers etc. one can compute for a knot K, such that if the invariants of K,K* are different then K,K* are different. Two such invariants are Khovanov homology KH(K), and symplectic Khovanov homology SKH(K), which is defined by SKH(K)=HF(L,L*) for Lagrangians L,L* in a symplectic manifold M defined using K. We aim to prove the Seidel-Smith Conjecture, that KH(K)=SKH(K). This will give new insight and methods of proof in knot theory.The fourth problem uses the new version of HF(L,L*) to strengthen results of Wehrheim-Woodward relating Lagrangian Floer theory in different symplectic manifolds M_1,M_2, using Lagrangian correspondences . It shows this relation is associative , that is, going from M_1 to M_2 to M_3 is the same as going from M_1 to M_3. Here working with immersed Lagrangians is important, but current results deal only with embedded Lagrangians.

大多数现代几何学研究的是某种空间。微分几何中考虑的空间称为流形，这种空间局部看起来像n维欧几里德空间，但整体上却有一个有趣的形状。如果流形是封闭的，没有边，则它是紧致的。甜甜圈的表面是一个紧凑的二维流形。流形M的子流形N是M的一个子集，它本身就是一个流形，通常比M的维度小。有两种类型：嵌入子流形和浸入子流形，它们不能彼此相交(交叉)。人们通常认为流形具有一些额外的几何结构，如黎曼度量，它告诉你流形中路径的长度，或者辛结构，它告诉你二维子流形的面积。辛流形是力学的数学公式的基础，也是许多经典物理的基础。他们本身也很有趣。数学家喜欢它们，因为它们是极少数具有无限维对称性的结构之一，这赋予了辛几何一种不同寻常的、完全全局的味道。拉格朗日子流形是辛流形的一种特殊的子流形。给定辛流形M的两个紧致嵌入的拉格朗日子流形L，L*，在一定条件下可以定义Floer上同调群Hf(L，L*)，它们粗略地说是有限维向量空间。这个定义非常困难。为此，在M上选择一个辅助复结构J，并计算M中具有L和L并的边界(边)的J-全纯二维圆盘D。关于HF(L，L*)，值得注意的是它与J的选择无关，并且通过在拉格朗日子流形之间移动L和L*也是不变的。它编码了一些关于拉格朗日子流形的神秘的、不平凡的信息，人们无法通过任何其他已知的方式获得。它是辛几何中的一个强有力的工具。在以前EPSRC资助的研究中，Pi和Akaho将HF(L，L*)的定义从嵌入的拉格朗日扩展到浸入的拉格朗日。PI还开发了新技术(仓西(上)同源)，将简化和简化HF(L，L*)的定义。这项建议将利用这些想法。我们将首先开发一个新的，更简单和更通用的HF(L，L*)配方，用于浸泡L，L*，使用PI的新技术。然后我们将这个新的公式应用到四个问题上。第一个问题将证明一个关于HF(L，L*)的猜想，当L，L*是超卡勒流形中的复拉格朗日函数时。重点是，新版本的高频(L，L*)将具有使证明比现有的高频定义(L，L*)容易得多的技术特征。第二和第三个问题涉及纽结理论：研究三维空间中的纽结(本质上是弦的环)。两个结K，K*是相同的，如果你可以变形K到K*而不切断线。计算两个结是否相同是一个困难的问题。数学家定义了纽结不变量、数等，人们可以计算纽结K，这样，如果K，K*的不变量不同，那么K，K*也不同。两个这样的不变量是Khovanov同调KH(K)和辛Khovanov同调SKH(K)，它是由Kh(K)=Hf(L，L*)定义的，对于用K定义的辛流形M中的拉格朗日L，L*。我们的目的是证明Seidel-Smith猜想，KH(K)=SKh(K)。这将为纽结理论提供新的见解和证明方法。第四个问题使用新版本的HF(L，L)来加强Wehheim-Woodward在不同辛流形M_1，M_2中利用拉格朗日对应联系拉格朗日Floer理论的结果。它表明这种关系是结合的，即从M_1到M_2再到M_3等同于从M_1到M_3。在这里，使用浸没拉格朗日是重要的，但目前的结果只涉及嵌入的拉格朗日。