Mathematical Sciences: Invariants for 3- and 4-Manifolds

9302526自1982年首席研究员Simon Donaldson和R.Fintushel应用规范理论研究同调3-球面和光滑4-流形以来，Fintushel一直在研究同调3-球面和光滑4-流形。目前的项目沿袭了这一传统，利用首席研究员和R.Fintushel最近的工作和Taubes最近的工作，Morgan，Mrowka和Ruberman的工作，以及Kronheimer和Mrowka的工作来确定给定光滑4-流形上保持其同胚型和改变其微分同胚型的那些运算。此外，对于不具有Donaldson多项式的流形，即具有Euler特征加签名不能被4整除的流形，将研究奇异的光滑光滑结构。本项目的主要目的是分类光滑的单连通4-流形，即局部模拟在4维空间上的对象。当然，相对论时空是这种流形的最著名的例子。虽然时空的精确整体结构尚不为人所知，但许多这样的流形可以描述为复多项式系统的解。直到最近才发现了不能这样描述的例子，因此立即将4-流形分类的希望破灭了。由于这些例子，对于这种分类可能采取的形式，没有明智的猜测。这是当前项目的根本目的是进一步研究这些和其他“奇异的4-流形”，并将它们放置在一个更大的和较少的特别框架中。***

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. ***