On the structure of the $RO(G)$-graded homotopy of $H\underline{M}$ for cyclic $p$-groups

TL;DR

This paper analyzes the $RO(G)$-graded homotopy Mackey functors of Eilenberg-MacLane spectra for cyclic $p$-groups, providing new structural insights, induction theorems, and explicit computations relevant to slice spectral sequences.

Contribution

It introduces orientation classes and a generalized gold relation, establishing isomorphism regions and computing key cones for homotopy Mackey functors of specific spectra.

Findings

Defined orientation classes $u_V$ for $Har{R}$ spectra.

Proved two induction theorems for homotopy Mackey functors.

Computed positive and negative cones of $Har{bZ}$ and $Har{bA}$.

Abstract

We study the structure of the $RO(G)$-graded homotopy Mackey functors of any Eilenberg-MacLane spectrum $H\underline{M}$ for $G$ a cyclic $p$-group. When $\underline{R}$ is a Green functor, we define orientation classes $u_V$ for $H\underline{R}$ and deduce a generalized gold relation. We deduce the $a_V,u_V$-isomorphism regions of the $RO(G)$-graded homotopy Mackey functors and prove two induction theorems. As applications, we compute the positive cone of $H\underline{\mathbb{A}}$, as well as the positive and negative cones of $H\underline{\mathbb{Z}}$. The latter two cones are essential to the slice spectral sequences of $MU^{((C_{2^n}))}$ and its variants.