In its simplest form, equivariant homotopy theory is the study of homotopy theory with the addition of actions by a group G. By mixing representation theory into homotopy theory, we create additional structure and complexities to consider. To every representation of our group, we may take the one point compactification and the group will then act on the resulting sphere. In so called “genuine” G-spectra, these representation spheres are invertible, so in particular we may grade cohomology theories on (virtual) G-representations. In this introductory talk, I will go over some of the additional properties and structures of equivariant homotopy theory and if time permits illustrate these structures with a cohomology computation.
