Topics in Topological Fixed Point Theory

拓扑不动点理论专题

• 批准号: 0805968
• 负责人: Peter Wong
• 金额: $ 10.72万
• 依托单位: Bates College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805968&HistoricalAwards=false
• 关键词: Topics; Topological; Fixed; Point; Theory

One of the central issues in fixed point theory is to compute the Nielsen number of a self map of a compact manifold X. This problem is closely related to understanding induced twisted conjugacy classes in the fundamental group of X. A related question is the coincidence problem of two maps from X to Y. A special case is a Borsuk-Ulam type problem. Wong proposes three sets of problems on (i) the study of the Reidemeister number; (ii) the study of the positive codimensional coincidence problem when the dimension of X is greater than that of Y ; and (iii) combinatorial approaches to Borsuk-Ulam type problems.Topological fixed point theory is a classical topic which generated much research in algebraic topology during the first half of the twentieth century. This proposal concerns the calculation of Nielsen numbers, an invariant in fixed point theory. Nielsen numbers have been shown to have applications in dynamical sysytems and nonlinear analysis. Wong proposes to study Nielsen numbers in some especially difficult situations where another more easily computed invariant, the Reidemister number, may differ from the Nielsen number. Combinatorial approaches to these calculations are also envisioned.

不动点理论的核心问题之一是计算紧致流形X上自映射的Nielsen数。这个问题与理解X的基本群中的诱导扭共轭类密切相关。一个相关的问题是两个从X到Y的映射的重合问题。一个特殊的情况是Borsuk-Ulam型问题。Wong提出了三组问题：（i）Reidemeister数的研究;（ii）当X的维数大于Y的维数时，正余维重合问题的研究;（iii）Borsuk-Ulam型问题的组合方法。拓扑不动点理论是一个经典的课题，在二十世纪上半叶的代数拓扑中产生了大量的研究。这个建议涉及到计算尼尔森数，不动点理论中的一个不变量。Nielsen数在动力系统和非线性分析中有着广泛的应用。 黄建议研究尼尔森数在一些特别困难的情况下，另一个更容易计算的不变量，Reidemister数，可能会不同于尼尔森数。这些计算的组合方法也被设想。