Decomposing C2-Equivariant Spectra

Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2, the cyclic group of order two.  A recent structure theorem for RO(C_2)-graded cohomology with Z/2 coefficients substantially simplifies computations.  The structure theorem says the cohomology of any finite C2-CW complex decomposes as a direct sum of two basic pieces: cohomologies of representation spheres and cohomologies of spheres with the antipodal action.  This decomposition lifts to a splitting at the spectrum level.  In joint work with Dan Dugger and Christy Hazel we extend this result to a classification of compact modules over the genuine equivariant Eilenberg-MacLane spectrum HZ/2.