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?

我们将研究三维和四维流形的对称有限群的存在性和分类。最近，基于W.瑟斯顿、M.Freedman和S.Donaldson开创的技术，我们对低维流形的理解有了巨大的增长。这些新的技术和结果为更系统地研究这些流形的对称性提供了工具，并显著地尖锐了人们可以希望解决的问题。4-流形的对称性的存在性：Freedman展示了如何根据它们的交集形式和稳定的三角剖分障碍来分类所有单连通的、闭拓扑4-流形。那么，我们应该问一问这些流形承认了什么对称性。主要调查者的结果表明，对于所有的奇数n，所有这些4-流形都存在局部线性的Z作用，并且除了可能的伪复射影平面外，所有的流形都存在不一定是局部线性的Z作用。所有上述作用都被构造为同调平凡的，从而导致在同调上的同一性。人们不禁要问：闭单连通拓扑4-流形的交形式的什么自同构可以由该流形的周期同胚来实现？