Topology of Smooth 4-Manifolds

提案：DMS-9971440主要研究者：Selman Akbulut摘要：主要研究者将研究光滑4-流形的拓扑结构。 通过将光滑流形分解为复杂的片段，他将使用复杂和辛流形的技术来理解这种分解对光滑4-流形拓扑结构的限制。 主要研究者希望将解决“Scharlemann问题”和Fintushel-Stern构造的技术应用于构造4维假s-配边的问题，以及构造“与圆相交的三个球体”的假副本的问题。他想研究的拓扑Horikawa表面，并希望给予肯定的答案，各种分支覆盖aesturtures。首席研究员打算研究广义四维空间（4流形），他希望找到新的。这些空间在物理学中自然出现：相对论（时空）和量子场论。 最近有令人兴奋的发展，这些空间的结构来自物理学的想法，但我们仍然不知道是否某些基本的奇异4-流形可能存在。如果它们存在，它们将是已知的基本构建块。我们想研究可能使它们存在的结构。

Proposal: DMS-9971440Principal Investigator: Selman AkbulutAbstract: The principal investigator will investigate the topology of smooth 4-manifolds. By decomposing smooth manifolds into complex pieces, he will use the techniques of complex and symplectic manifolds to understand the restriction this decomposition imposes on the topology of smooth 4-manifolds. The principal investigator hopes to apply the techniques from the solution of the `Scharlemann problem' and Fintushel-Stern constructions to the problem of constructing 4-dimensional fake s-cobordisms, and to the problem of constructing a fake copy of `three-sphere crossed with circle'. He would like to study topology of Horikawa surfaces and hopes to give affirmative answers to various branched covering conjectures.The principal investigator intends to study generalized 4-dimensional spaces (4-manifolds); he hopes to find new ones. These spaces arise naturally in physics: relativity (space-time) and quantum field theories. Recently there have been exciting developments about the structure of these spaces coming from the ideas of physics, but we still don't know if certain basic exotic 4-manifolds could exist. If they exist they will be basic building blocks of the known ones. We would like to investigate possible constructions which could give their existence.