{
  "id": "2309.17242",
  "version": "v1",
  "published": "2023-09-29T13:51:14.000Z",
  "updated": "2023-09-29T13:51:14.000Z",
  "title": "On the structure of the $RO(G)$-graded homotopy of $H\\M$ for cyclic $p$-groups",
  "authors": [
    "Igor Sikora",
    "Guoqi Yan"
  ],
  "comment": "33 pages. Initial version, comments more than welcome!",
  "categories": [
    "math.AT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-29T13:51:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q91",
      "55P91"
    ],
    "keywords": [
      "graded homotopy mackey functors",
      "slice spectral sequences",
      "define orientation classes",
      "green functor",
      "induction theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}