Borel equivariant homotopy theory

对称性的研究是数学中最古老的话题之一。在代数拓扑学中，人们通过寻找将空间变换到自身上的运动来研究拓扑空间的对称性。旋转的篮球是将球体转变为自身的运动的一个例子。球体是所有空间中最对称的，它们从各个方向看起来都是一样的，从拓扑学开始就受到了严格的审查。用正交矩阵可以很容易地理解球体的刚性运动。光滑运动和连续运动更难研究，但一些结果是已知的。拓扑空间最基本的对称性是同伦对称性，它允许任意伸展和折叠的运动，但不允许撕裂。格罗达尔和史密斯一直在研究球体的同伦运动，令我们惊讶的是，我们现在对球体的同伦对称性有了完全的理解，这比人们对光滑运动或连续运动的理解要好得多。这个项目的目的是推广Grodal-Smith关于球面同伦对称性的结果，得到关于其他流形和有限复形的同伦对称性的结果。

The study of symmetry is one of oldest topics in mathematics. In Algebraic Topology, one studies the symmetry of topological spaces by looking for motions that transform the space onto itself. A spinning basketball is an example of motions that transform the sphere onto itself. Spheres are the most symmetric of all spaces, they look the same from all directions, and have received intense scrutiny since the beginning of topology. The rigid motions of a sphere are easily understood using orthogonal matrices. Smooth motions and continuous motions are harder to study but some results are known. The most fundamental symmetry of a topological space is homotopy symmetry, allowing motions with arbitrary stretching and collapsing but no tearing. Grodal and Smith have been studying the homotopy motions of spheres and much to our surprise we now have a complete understanding of the homotopy symmetry of a sphere, a much better understanding than one has for smooth motions or for continuous motions. The goal of this project is to extend the Grodal-Smith results about the homotopy symmetry of spheres to obtain results about the homotopy symmetry of other manifolds and finite complexes.