On the regular slice spectral sequence

Abstract

In this thesis, we analyze a variant of the slice spectral sequence of [HHR (or SSS) called the regular slice spectral sequence (or RSSS). This latter spectral sequence is defined using only the regular slice cells. We show that the regular slice tower of a spectrum is just the suspension of the slice tower of the desuspension of that spectrum. Hence, many results for the RSSS are equivalent to corresponding results for the SSS. However, the RSSS has many multiplicative properties that the SSS lacks. Also, the slice towers that have been computed prior to this thesis happen to coincide with the corresponding regular slice towers. Hence, we find the RSSS to be much better behaved than the SSS. We give a comprehensive study of its basic properties, including multiplicative structure, Toda brackets, interaction with the norm functor of [HHRJ, vanishing lines and preservation of various kinds of extra structure. We identify a large portion of the first page of the spectral sequence algebraically by relating the RSSS to the homotopy orbit and homotopy fixed point spectral sequences, and determine the edge homomorphisms. We also give formulas for the slice towers of various families of spectra, and give several sample computations. The regular slice tower for equivariant complex K-theory is used to prove a special case of the Atiyah-Segal completion theorem. We also prove two conjectures of Hill from [Hill concerning the slice towers of Eilenberg MacLane spectra, as well as spectra that are concentrated over a normal subgroup.