A new formulation of the equivariant slice filtration with applications to C_p-slices

This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a G-spectrum is slice n-connective. In particular, we show that a G-spectrum is slice greater than or equal to n if and only if for all subgroups H, the H-geometric fixed points are (n/|H| − 1)-connected. We use this to determine when smashing with a virtual representation sphere S^V induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary C_p-spectrum and show how a very small number of functors determine all of the slices for C_pn-spectra.