The Structure of RO(G)-Graded Homotopy of Eilenberg-MacLane Spectra for Cyclic Two-Groups and the Slice Spectral Sequences

We study the RO(G)-graded homotopy Mackey functors of Eilenberg-Mac Lane spectra for cyclic p-groups. One innovation is the use of the generalized Tate squares introduced by Greenlees-May in the computations. We exploit the power of these generalized Tate squares further by applying them to the study of the equivariant slice spectral sequence invented by Dugger which is later generalized by Hill-Hopkins-Ravenel in their solution of the Kervaire invariant problem. The Tate squares for different families provide stratification of the slice spectral sequences. We deduce vanishing lines and transchromatic phenomenon in the negative cones of these spectral sequences, extending the work of Meier-Shi-Zeng on the positive cones.

We also compute RO(G)-graded coefficients in some other cases, as illustrations of the usefulness of the Tate squares in equivariant computations, especially when dealing with the multiplicative structures.