Three and Four-Manifolds

摘要：本文研究了四维流形光滑拓扑中的几个问题。一种是找到theseberg - witten不变量的拓扑定义，也许是通过利用4流形上的几乎复杂的结构，这些结构在某种意义上是“紧的”，而不是辛的。其次是找到一个以给定封闭3-流形的三个副本（方向交替）的连通和为边界的4-流形的自然构造。第三（与Paul Melvin）完成某些复杂椭圆曲面的车体结构研究。四维流形是所有维度中最不为人所知的，但它们经常出现，例如，与物理时空的三个空间维度和一个时间维度有关。基本问题是对这些流形进行分类，分类意味着用相对简单的可以计算的不变量来理解它们。1994年数学物理学家发现的Seiberg-Witten不变量是目前最好的不变量集，在理解和扩展它们方面需要做很多工作。一种技术是柄体理论，它涉及到用简单的构件来描述流形的简单方法。

Proposal: DMS-9975171PI: Robion KirbyAbstract: This proposal concerns several problems in the smooth topology of fourdimensional manifolds. One is to find a topological definition of theSeiberg-Witten invariants, perhaps by exploiting almost complexstructures on 4-manifolds which are "tight" in some sense other thanbeing symplectic. Second is to find a natural construction of a4-manifold which has as boundary the connected sum of three copies(with alternating orientations) of a given closed 3-manifold. Thirdis (with Paul Melvin) to finish work on the handlebody structure ofcertain complex elliptic surfaces.Manifolds of dimension four are the least understood of alldimensions, yet they occur frequently, for example, in relation to thethree spatial and one temporal dimensions of physical space-time. Afundamental problem is to classify these manifolds, where classifymeans to understand them in terms of relatively simple invariantswhich can be computed. The Seiberg-Witten invariants which werediscovered in 1994 by mathematical physicists are the best current setof invariants, and much work needs to be done in understanding andextending them. One technique is handlebody theory which involves asimple way of describing a manifold in terms of simple buildingblocks.