Mathematical Sciences: Geometric and Algebraic Topology

数学科学：几何和代数拓扑

• 批准号: 9626111
• 负责人: Allen Hatcher
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626111&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; Algebraic; Topology

9626111 Hatcher The two investigators will continue their research into homological properties of automorphism groups of free groups, in particular, to determine whether there is nontrivial rational homology in the stable dimension range and whether these automorphism groups are virtual duality groups. The homotopy type of related complexes on which the automorphism groups act, for example the complex of free factors of a free group, will also be investigated. Another problem which will be studied is to determine the homotopy type of the space of all embeddings of a link in 3-space, extending known results for knots. In particular, it should be possible to answer Kontsevich's question whether this space has finite homotopy type. In more general terms, this involves determining all the essentially different ways of removing unnecessary tangles from a knotted piece of string or from several pieces which are knotted and linked together. With regard to the other part of the project, automorphisms of free groups play a fundamental role in group theory (the general study of symmetry) somewhat analogous to matrices in linear algebra, and the goal is to understand just how close this parallel is: When one takes a geometric view of both sorts of algebraic objects, is the picture qualitatively the same, or qualitatively different? ***

小行星9626111 两位研究者将继续研究自由群的自同构群的同调性质，特别是确定在稳定维数范围内是否存在非平凡有理同调，以及这些自同构群是否是虚对偶群。 同伦类型的相关复杂的自同构群的行为，例如复杂的自由因子的自由群，也将被调查。 另一个问题，将研究的是，以确定同伦类型的空间的所有嵌入的链接在3-空间，扩展已知的结果结。 特别是，它应该有可能回答Kontsevich的问题，这个空间是否有有限同伦型。 更一般地说，这涉及确定从打结的一段绳子或从打结并连接在一起的几段绳子上去除不必要的缠结的所有基本上不同的方法。 关于项目的另一部分，自由群的自同构在群论（对称性的一般研究）中起着基本的作用，有点类似于线性代数中的矩阵，目标是理解这种相似性有多接近：当一个人对两种代数对象进行几何观察时，图片是定性相同的，还是定性不同的？ ***