Cohomology operations from S¹-cobordisms in Floer homology
Cohomology operations from S¹-cobordisms in Floer homology
复制标题
弗洛尔同调中 S1-配边的上同调运算
DOI：
10.3929/ethz-a-001475365
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发表时间：
1995
影响因子：
1.8
通讯作者：
M. Schwarz
中科院分区：
文献类型：
--
作者：
M. Schwarz
本文将Floer同调理论作为辛流形（M，to）的loop空间上辛作用泛函的相对莫尔斯理论.假设M是闭的，并且上同调类{w}，ci（TM）H2{M）在^（M）上消失。在Floer同调中，利用一类非线性Fredholm算子分析了无限标准柱1 × 5 '到M的映射的Hamilton作用泛函的相对梯度流.所涉及的算子是Cauchy-Riemann型椭圆型偏微分算子。边界条件由固定Hamilton方程的可收缩非退化1周期解给出。结果表明，这些非线性Cauchy-Riemann算子的Floer同调的解析概念可以从标准柱面推广到任意的Riemann曲面上，这些曲面要么是封闭的，要么是具有标准柱面结构和特定方向的端点.一个非线性Fredholm分析被用来定义代数运算的Floer同调群。在我们的工作中，作为一个上同调理论的等价描述被选择。在特定的条件下，已知这些群与M作为分次Z2-向量空间的奇异同调正则同构。与黎曼曲面相关的上同调运算在这个分次向量空间上产生了进一步的代数结构。主要结果是：Floer上同调HF*（M，1 j2）具有Z2上Z-分次结合交换代数的结构，该代数具有单位元和非退化对称双线性型.通过计算一般条件下非线性Cauchy-Riemann型偏微分方程的解，定义了Riemann曲面S上Floer上同调的运算Z（Ti）.证明了算子Z（E）由模型曲面S的拓扑类型（即紧化曲面的有向同胚类）唯一确定。这里，定向的圆柱形端被紧化为（0，1] x S1和[-1，0] x S1，respectively。因此，每一个这样的曲面都可以被看作是一个有向的51配边。结果表明，我们的理论导致一个函子Z，它为每个这样的具有指定方向的a + B端的配边S分配一个多线性算子Z（T，）：®hHF*。证明了这个函子Z满足Atiyah [3]意义下的1 + 1维拓扑场论的公理。这等价于将向量空间HF* 描述为Z2上的代数，其具有与乘法相容的单位和非退化双线性形式。
In this work, Floer homology is considered as a relative Morse theory for the symplectic action functional on the loop space of a symplectic manifold (M, to). It is assumed that M is closed and the cohomology classes {w},ci(TM) H2{M) vanish on ^(M). In Floer homology the relative gradient flow for the Hamiltonian action functional is analysed by means of a class of nonlinear Fredholm operators for maps of the infinite standard cylinder 1x5' into M. The operators involved are elliptic partial differential operators of Cauchy-Riemann type. The boundary conditions are given by contractible non-degenerate 1periodic solutions of a fixed Hamiltonian equation. It is shown that the analytical concept of Floer homology for these nonlin¬ ear Cauchy-Riemann operators can be generalized from the standard cylinder to arbitrary Riemann surfaces which are either closed or have ends which are endowed with the standard cylindrical structure and carry a specified orien¬ tation. A nonlinear Fredholm analysis is used to define algebraic operations on the Floer homology groups. In our work the equivalent description as a cohomology theory is chosen. Under the specified conditions these groups are known to be canonically isomorphic to the singular homology of M as a graded Z2-vector space. The cohomological operations associated to Riemann surfaces give rise to further algebraic structures on this graded vector space. The main result is that the Floer cohomology HF*{M,1j2) carries the struc¬ ture of a Z-graded associative and commutative algebra over Z2 with unit and with a non-degenerate symmetric bilinear form. The operation Z(Ti) on Floer cohomology associated to a Riemann surface S is defined by counting solutions of nonlinear Cauchy-Riemann type partial differential equations under generic conditions. It is proven that the operator Z(E) is uniquely determined by the topological type of the model surface S, that is, the oriented homeomorphisms class of the compactified surface. Here, the oriented cylindrical ends are compactified as (0,1] x S1 and [— 1,0) x S1, re¬ spectively. Thus, every such surface can be viewed as an oriented 51-cobordism. It turns out that our theory leads to a functor Z which assigns to each such cobordism S with a + b ends of specified orientation a multi-linear operator Z{T,): ®hHF*. It is proven that this functor Z satisfies the ax¬ ioms of a topological field theory in dimension 1 + 1 in the sense of Atiyah, [3]. This is equivalent to describing the vector space HF* as an algebra over Z2 with unit and non-degenerate bilinear form compatible with the multiplication.