[论文解读] Generation for Lagrangian cobordisms in Weinstein manifolds

Hiro TANAKA|arXiv (Cornell University)|Oct 24, 2018

Homotopy and Cohomology in Algebraic Topology参考文献 11被引用 7

一句话总结

该论文证明了在停止的Weinstein流形中，Lagrangian余核与连接圆盘生成Lagrangian cobordism的$∞$-范畴，通过cobordism到标准对象提供了任意精确bran的构造性几何实现。结果表明，每个稳定化的bran都存在Lagrangian cobordism至余切纤维的不相交并集与零对象，提供了强于以往Fukaya范畴生成结果的几何实现。

ABSTRACT

We prove that Lagrangian cocores and Lagrangian linking disks of a stopped Weinstein manifold generate the Lagrangian cobordism infinity-category. As a geometric consequence, we see that any brane (after stabilization) admits a Lagrangian cobordism to a disjoint union of some standard collection of branes (cocores, linking disks, and a zero object). For example, when our stopped Weinstein manifold is a point stopped by itself, we find that any exact brane in Euclidean space admits a Lagrangian cobordism to a disjoint union of cotangent fibers and a zero object. (This is a stronger statement than one could obtain from purely Fukaya-categorical generation results.) Our methods are constructive. For example, when our Weinstein manifold is a point, after stabilization we can resolve the conormal to a compact manifold A of R^n by a sequence of cotangent fibers; the resulting filtration realizes, after passage to the wrapped Fukaya category, the Morse cochain complex of A associated to (and hence filtered by) a generic ``distance to a point'' function; the associated gradeds are the reduced homologies of the Morse attaching spheres. There is also an algebraic consequence. Lagrangian cobordism theory is conjectured (in analogue to classical cobordism theory) to be linear over a ring spectrum L controlling Lagrangian cobordisms between cotangent fibers in Euclidean spaces. Our main theorem gives strong evidence for this conjecture: The infinity-category of Lagrangians and their cobordisms in R^infinity is equivalent to a full subcategory of modules over L. We conclude by proving a π_0-level theorem that gives further evidence of the above conjecture: We exhibit a π_0-level symmetric monoidal structure compatible with the linear structure of L-modules.

研究动机与目标

• 建立停止的Weinstein流形中Lagrangian cobordism的$∞$-范畴的几何生成结果。
• 为任意精确bran提供一种构造性方法，将其表示为至标准对象集合（包括余核、连接圆盘和零对象）的Lagrangian cobordism。
• 通过显式构造在无穷远处行为受控的cobordism，提供强于纯Fukaya范畴生成的几何实现。
• 为Lagrangian cobordism理论在控制欧氏空间中余切纤维之间cobordism的环谱$σ$上的线性性提供代数证据。
• 在Lagrangian cobordism的同伦范畴上构造$π_0$-层级的对称张量结构，与$σ$-模的线性结构相容。

提出的方法

• 证明通过哈密顿同伦与锥结构显式构造Lagrangian cobordism，确保无穷远处的势函数为零。
• 该方法依赖于通过一系列余切纤维解析紧致子流形$A \subset \mathbb{R}^n$的法丛，从而得到实现$A$的Morse上链复的滤过结构。
• 构造利用了卷绕Fukaya范畴，将相应级数对象解释为Morse附着球面的约化同调。
• 作者通过笛卡尔积与交换映射在$σ$-模的同伦范畴上定义对称张量结构，并利用旋转向量场验证。
• cobordism范畴中的映射锥通过与映射的锥的积几何实现，保持三角结构。
• 通过与Lagrangian子流形的张量积定义$σ$-模在cobordism范畴上的双模作用，且与对称张量结构相容。

实验结果

研究问题

• RQ1 在停止的Weinstein流形中，每个精确bran是否可通过Lagrangian cobordism几何实现为至标准对象集合？
• RQ2 停止的Weinstein流形中Lagrangian cobordism的$∞$-范畴是否由余核与连接圆盘几何生成？
• RQ3 Lagrangian cobordism理论在环谱$σ$上为线性这一代数猜想，是否可通过范畴等价性得到支持？
• RQ4 在Lagrangian cobordism的同伦范畴上是否存在与$σ$-模结构相容的$π_0$-层级对称张量结构？
• RQ5 通过余切纤维对法丛的滤过结构如何与底流形的Morse上链复相关联？

主要发现

• 在停止的Weinstein流形中，Lagrangian cobordism的$∞$-范畴作为稳定$∞$-范畴，由Lagrangian余核与连接圆盘生成。
• 在停止的Weinstein流形中，每个稳定化的精确bran都存在Lagrangian cobordism至余核、连接圆盘与零对象的不相交并集。
• 当Weinstein流形为一点时，$T^*ℝ^N$中的每个精确bran都存在cobordism至余切纤维与零对象的不相交并集。
• 该构造通过余切纤维的滤过实现紧致子流形$A \subset \mathbb{R}^n$的Morse上链复，其相应级数部分同构于Morse附着球面的约化同调。
• $ℝ^∞$中Lagrangian cobordism的$∞$-范畴与环谱$σ$的模的一个全子范畴等价。
• 在$σ$-模的同伦范畴上构造了与双模作用和映射锥结构相容的$π_0$-层级对称张量结构。