The moduli space of homotopy G-spheres

I will discuss joint work with Jesper Grodal. A homotopy G sphere is a space that is homotopy equivalent to a sphere and has an action of the group G. Two homotopy G spheres are equivalent if there is a zigzag of equivariant weak equivalneces that connects them. We classify homotopy G spheres for all finite groups. We compute the monoid of homotopy classes of self maps of each homotopy G sphere and we show that most of the higher homotopy groups of the space of self maps finite.