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

Igor Sikora, Guoqi Yan

Published 2023-09-29Version 1

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