Mathematical Sciences: Topology of Smooth 4-Manifolds

数学科学：光滑 4 流形拓扑

• 批准号: 9626204
• 负责人: Selman Akbulut
• 金额: $ 3.82万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626204&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Smooth; Manifolds

9626204 Akbulut This project has two parts, (a)topology of smooth 4-manifolds and (b)topology of real algebraic sets. In (a) the investigator intends to construct some new 4-manifolds (such as manifolds with b+ even, and a product of spheres) by using the new Seiberg-Witten technology and by some new algebraic topological techniques (torsion invariants for 4-manifolds). He would also like to use symplectic/contact structures to get a better structure theorem for smooth manifolds (i.e., generalized "cork" structures). In (b) he intends to continue with the "topological classification program" previously initiated jointly with H.King. He will also try to construct submanifolds of Euclidean space that cannot be isotopic to a nonsingular algebraic subset (i.e., strongly transcendental manifolds). The investigator would like to construct some new exotic 4-dimensional manifolds (generalizations of 4-dimensional spaces) that have not been detected by the current techniques. He also would like to classify "algebraic spaces" (spaces that are described by systems of polynomial equations). The mysteries of 4-dimensional geometry and topology have been recognized as of more than purely theoretical interest ever since the acceptance of time as a fourth physical dimension, e.g., in Einstein's theory of special relativity in the early years of this century. ***

9626204 Akbulut 该项目有两个部分，(a) 光滑 4 流形拓扑和 (b) 实代数集拓扑。 在(a)中，研究者打算通过使用新的Seiberg-Witten技术和一些新的代数拓扑技术（4流形的扭转不变量）来构造一些新的4流形（例如b+偶数的流形和球体的乘积）。 他还想使用辛/接触结构来获得更好的光滑流形结构定理（即广义的“软木塞”结构）。 在(b)中，他打算继续之前与H.King联合发起的“拓扑分类计划”。 他还将尝试构造欧几里得空间的子流形，该子流形不能与非奇异代数子集同位素（即强超越流形）。 研究人员想要构建一些当前技术尚未检测到的新的奇异 4 维流形（4 维空间的概括）。 他还想对“代数空间”（由多项式方程组描述的空间）进行分类。 自从接受时间作为第四个物理维度以来，例如本世纪初爱因斯坦的狭义相对论，四维几何和拓扑的奥秘就被认为不仅仅是纯粹的理论兴趣。 ***