\(RO(C_3)\)-graded cohomology

For spaces with an action by a group \(G\), one can compute an equivariant analogue of singular cohomology referred to as \(RO(G)\)-graded Bredon cohomology.  Instead of coefficients in an abelian group, this theory requires coefficients in a Mackey functor.  Computations in this setting are often challenging and not well understood, even for \(G = C_p\), the cyclic group of order \(p\). In this talk, I will briefly introduce \(RO(G)\)-graded cohomology and discuss some results toward a structure theorem for \(RO(C_3)\)-graded cohomology with \(\underline{\mathbb{Z}/3}\)-coefficients.  The structure theorem would describe the building blocks for the cohomology of all finite \(C_3\)-spaces.  A recent structure theorem for \(C_2\) with \(\underline{\mathbb{Z}/2}\)-coefficients shows the building blocks depend on two types of spheres, representation spheres and antipodal spheres.  For \(C_3\), we will see that we need two types of spheres, as well as a new space that is not a sphere at all.