Mathematical Sciences: Invariants for 3- and 4-Manifolds

小行星9302526 自1982年西蒙唐纳森的工作以来，首席研究员和R。Fintushel应用规范理论研究了同调3-球面和光滑4-流形。 目前的项目遵循这一传统，利用主要研究者和R。Fintushel和最近的工作Taubes，工作的摩根，Mrowka，和Ruberman，以及工作的Kronheimer和Mrowka，以确定这些行动对一个给定的光滑4流形保持其同胚类型和改变其同胚类型。 此外，奇异光滑结构将被调查的那些流形，没有唐纳森多项式，即流形的欧拉特征加上签名不能被4整除。 该项目的主要目标是对光滑单连通4-流形进行分类，即在四维空间上局部建模的对象。 当然，相对论性时空是这种流形最著名的例子。 虽然时空的精确整体结构还不清楚，但许多这样的流形可以被描述为复多项式系统的解。 直到最近才发现了一些不能如此描述的例子，从而使对4-流形进行分类的希望破灭。 因为有了这些例子，所以对于这种分类可能采取什么形式，没有什么明智的猜测。 当前项目的基本目的是进一步研究这些和其他“奇异的4-流形”，并将它们放在一个更大和更少的特设框架中。 ***

9302526 Fintushel Since the 1982 work of Simon Donaldson, the principal investigator and R. Fintushel have applied gauge theory to study homology 3-spheres and smooth 4-manifolds. The current project follows in this tradition, utilizing the recent work of the principal investigator and R. Fintushel and the recent work of Taubes, the work of Morgan, Mrowka, and Ruberman, and the work of Kronheimer and Mrowka to determine those operations on a given smooth 4-manifold which preserve its homeomorphism type and alter its diffeomorphism type. Further, exotic smooth structures will be investigated for those manifolds which have no Donaldson polynomials, i.e. manifolds with Euler characteristic plus signature not divisible by 4. The major thrust of this project is to classify smooth simply-connected 4-manifolds, i.e. objects that are locally modeled on 4-dimensional space. Relativistic space-time is, of course, the best known example of such a manifold. Although the precise global structure of space-time is not known, many such manifolds can be described as solutions to systems of complex polynomials. Only recently have there been discovered examples which cannot be so described, thus dashing immediate hopes of classifying 4-manifolds. Because of these examples, there is no intelligent conjecture as to what form this classification might take. It is the fundamental purpose of the current project to investigate these and other "exotic 4-manifolds" further and to place them in a larger and less ad hoc framework. ***