Mathematical Sciences: Symmetries of Low-Dimensional Manifolds

数学科学：低维流形的对称性

• 批准号: 9002779
• 负责人: Allan Edmonds
• 金额: $ 1.56万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-11-15 至 1995-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002779&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetries; Low; Dimensional

The existence and classification of finite groups of symmetries of manifolds of dimensions three and four will be studied. Recently there has been a tremendous growth in our understanding of low-dimensional manifolds, based on techniques pioneered by W. Thurston, M. Freedman, and S. Donaldson. These new techniques and results have provided the tools to investigate much more systematically the symmetries of these manifolds and to sharpen significantly the questions that one can hope to address. Existence of symmetries of 4-manifolds: Freedman showed how to classify all simply connected, closed topological 4-manifolds in terms of their intersection forms and stable triangulation obstructions. It is then appropriate to ask what symmetries these manifolds admit. Results of the Principal Investigator show that all these 4-manifolds admit locally linear Zn actions for all odd n and that all except possibly the fake complex projective plane admit Z actions that are not necessarily locally linear. All the actions above were constructed so as to be homologically trivial, inducing the identity on homology. One is led to ask: What automorphisms of the intersection form of a closed simply connected topological 4-manifold can be realized by a periodic homeomorphism of the manifold?

有限群的存在性和分类 三维和四维流形的对称性 研究了 最近，我们的业务有了巨大的增长， 低维流形的理解，基于技术 由W. Thurston，M. Freedman和S.唐纳森。 这些 新的技术和结果提供了研究的工具 更系统地研究这些流形的对称性， 使问题更加尖锐， 地址. 存在的对称性的4-流形：弗里德曼表明如何 分类所有单连通闭拓扑4-流形 从相交形式和稳定三角剖分的角度 障碍物 那么我们就可以问， 这些流形承认。 主要研究者的结果 证明了所有这些4-流形都允许局部线性Zn作用 对于所有奇数n，除了假复形， 射影平面允许Z作用， 局部线性 上述所有行动都是为了 同调平凡的，从而导出同调上的恒等式。 一是 这就引出了一个问题：a的交集形式的自同构是什么？ 可以实现闭单连通拓扑4-流形 流形的周期同胚