Fukaya categories of the torus and Dehn surgery
Fukaya categories of the torus and Dehn surgery
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DOI：
10.1073/pnas.1018918108
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发表时间：
2011-05-17
影响因子：
11.1
通讯作者：
Perutz, Timothy
中科院分区：
文献类型：
--
作者：
Lekili, Yanki;Perutz, Timothy
本文是Heegaard Floer理论从闭3-流形到具有两个边界分支的紧致3-流形上同调推广工作的一部分。我们描述了这一理论的第一个有趣的例子：边界S-2余积T-2的3-流形的不变量被认为是被穿孔的2-环面的Fukaya范畴上的模。我们给出了Heegaard Floer同调中Dehn手术三角的正确性的一个简短证明。我们证明了由2-环面的Fukaya范畴中两个基本对象的上同调构成的分次代数A上的A(无穷)-结构仅由两个参数(m(6)，m(8))决定，这两个参数取自A的Hochschild上同调.对于Fukaya范畴本身，m(6)不等于0.
This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S-2 coproduct T-2, regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A(infinity)-structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category of the punctured 2-torus are governed by just two parameters (m(6), m(8)), extracted from the Hochschild cohomology of A. For the Fukaya category itself, m(6) not equal 0.