Topology of Smooth 4-Manifolds

光滑4流形拓扑

• 批准号: 0204579
• 负责人: Selman Akbulut
• 金额: $ 21.32万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204579&HistoricalAwards=false
• 关键词: Topology; Smooth; Manifolds

DMS-0204579Selman AkbulutProposer plans to investigate the topology of smooth 4-manifolds by decomposing them into Stein pieces which are easy to understand (since they turn out to be Lefschetz fibrations over the 2-disk). He would like use the techniques of complex and symplectic manifolds to understand the restriction this decomposition imposes to the topology of the original manifold. Proposer hopes to apply these techniques to attack some unsolved problems of 4-dimensional topology, such as finding 4-dimensional fake s-cobordisms, and to the problem of constructing a fake S^3 x S^1. Proposer also plans to work on the topological problems arising from complex algebraic geometry, such as problems coming from the complex conjugation, branched covers and the curve counting problem in real algebraic geometry.Proposer plans to investigate structure of 4-dimensional manifolds by decomposing them into pieces that are topologically easier to understand (4-manifolds are spaces which locally look like 4-dimensional Euclidean space-time we live in). These smaller pieces carry certain `natural complex structures' which analytical techniques apply. By these techniques Principal investigator hopes to construct certain 4-manifolds believed to exists but not yet have been found. One of the reasons 4-manifolds are of interest is because of the physicists model of the 10-dimensional universe which consists of a 4-dimensional space-time plus 6-dimensional complex piece.

DMS-0204579Selman AkbulutProposer 计划通过将光滑 4 流形分解为易于理解的斯坦因片来研究其拓扑（因为它们是 2 盘上的 Lefschetz 纤维）。他希望使用复流形和辛流形的技术来理解这种分解对原始流形拓扑的限制。提案者希望应用这些技术来解决 4 维拓扑中一些未解决的问题，例如寻找 4 维假 s-配边，以及构造假 S^3 x S^1 的问题。提案者还计划研究复杂代数几何中产生的拓扑问题，例如复共轭、分支覆盖以及实代数几何中的曲线计数问题。提案者计划通过将4维流形分解成拓扑上更容易理解的片段来研究4维流形的结构（4维流形是局部看起来像4维欧几里德时空的空间） 我们住在）。这些较小的碎片带有某些应用分析技术的“天然复杂结构”。首席研究员希望通过这些技术构建某些据信存在但尚未发现的 4 流形。 4 流形令人感兴趣的原因之一是物理学家的 10 维宇宙模型，该模型由 4 维时空加上 6 维复杂部分组成。