Borel equivariant homotopy theory

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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.