A General Fredholm Theory and Applications
A General Fredholm Theory and Applications
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Fredholm 的一般理论和应用
DOI：
10.4310/cdm.2004.v2004.n1.a1
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
H. Hofer
中科院分区：
文献类型：
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作者：
H. Hofer
这里描述的理论是试图找到一个一般的抽象框架，在这个框架中，各种理论，如Gromov-Witten理论(GW)、Floer理论(FT)、接触同调(CH)和更一般的辛场论(SFT)可以从一般的观点来理解。让我们用一种有点过于简单化的形式来描述一下总体情况。一个共同的特征(除了GW，它具有较少的结构)是这样一个事实，我们在具有边角的空间上定义了无限多不同的Fredholm型问题，其中边界层可以用其他问题(在列表上)的乘积(或更一般的纤维乘积)来解释。在过于简化的形式中，解集是某个丛τ：Y→X的一段f的零点，其中该空间有一个边界∂X，并且存在一个配方(甚至许多配方)来从f=0的两个给定解x和x构造一个新的解，例如乘积x=x◦x。构造新的解的配方即使对于非解也是定义的，而∂X恰恰是作为乘积的点的空间。因此，我们有
The theory described here results from an attempt to find a general abstract framework in which various theories, like Gromov-Witten Theory (GW), Floer Theory (FT), Contact Homology (CH) and more generally Symplectic Field Theory (SFT) can be understood from a general point of view. Let us describe the general landscape in a somewhat oversimplified form. The common feature (with the exception of GW which has less structure) is the fact that we have infinitely many different Fredholm problems defined on spaces with boundary with corners, where the boundary strata can be explained in terms of products (or more generally fibered products) of other problems (on the list). In oversimplified form, the solution sets are zeros of a section f of some bundle τ : Y → X, where the space has a boundary ∂X, and where moreover there exists a recipe (or even many recipes) to construct from two given solutions x and x of f = 0 a new solution, say the product, x = x ◦x. The recipes for constructing new solutions are defined even for non-solutions and ∂X is precisely the space of points which are products. Hence we have