The Klein four homotopy Mackey functor structure of HF2

Abstract

Let G = C2 × C2 be the Klein four group. Partial computations of the RO(G)- graded homotopy Mackey functors π⋆HF2 of the equivariant Eilenberg-MacLane spectrum corresponding to the constant Mackey functor F2 can be found in the literature. In particular, Holler-Kriz computed in [2] the complete additive structure of the top levels π⋆GHF2 of the Mackey functors, and Guillou-Yarnall in [3] computed the homotopy Mackey functors graded by multiples of the regular representation ρ, namely the integer graded homotopy Mackey functors π∗(ΣkρHF2) for each k ∈ Z. In this thesis, we discuss the multiplicative structure of the top level π⋆GHF2 and moreover give a complete algebraic description of the homotopy Mackey functors making up π⋆HF2 graded by all of RO(G). Finally, we use the Bockstein spectral sequence to compute the portion of π⋆GHZ graded by actual representations using our algebraic description of π⋆GHF2. Show more