Three and Four-Manifolds

提案：DMS-9975171 PI：Robion Kirby摘要：该提案涉及四维流形光滑拓扑中的几个问题。 一个是找到一个拓扑定义的theSeiberg-Witten不变量，也许利用几乎复杂的结构4流形是“紧”在某种意义上以外的辛。 第二个问题是找到一个以给定闭3-流形的三个副本（交替定向）的连通和为边界的4-流形的自然构造。 第三个是（与保罗·梅尔文）完成了某些复杂椭圆曲面的非线性体结构的工作。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。 一个基本的问题是对这些流形进行分类，分类意味着用相对简单的不变量来理解它们，这些不变量可以计算。 数学物理学家于1994年发现的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还有许多工作要做。 其中一种技术是三体理论，它涉及一种简单的方法来描述一个简单的积木方面的流形。

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.