Mathematical Sciences: Geometric and Algebraic Topology

9626111 Hatcher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的理性同调，以及这些自同构群是否是虚对偶群。自同构群作用的相关配合物的同伦类型，例如自由群的自由因子的复形，也将被研究。另一个将要研究的问题是确定3空间中连杆的所有嵌入空间的同伦类型，扩展已知结的结果。特别地，应该可以回答Kontsevich的问题，这个空间是否具有有限同伦类型。更一般地说，这包括确定所有本质上不同的方法，以消除不必要的缠结，从一根打结的绳子或从几个打结和连接在一起的绳子上。关于项目的另一部分，自由群的自同构在群论（关于对称性的一般研究）中扮演着基本的角色，有点类似于线性代数中的矩阵，目标是理解这种平行有多接近：当一个人从几何角度看待这两种代数对象时，图像在性质上是相同的，还是在性质上不同？＊＊＊

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? ***