On the structure of the $RO(G)$-graded homotopy of $H\underline{M}$ for cyclic $p$-groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918145931870208 |
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| author | Sikora, Igor Yan, Guoqi |
| author_facet | Sikora, Igor Yan, Guoqi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_17242 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the structure of the $RO(G)$-graded homotopy of $H\underline{M}$ for cyclic $p$-groups Sikora, Igor Yan, Guoqi Algebraic Topology 55Q91 (Primary), 55P91 (Secondary) 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. |
| title | On the structure of the $RO(G)$-graded homotopy of $H\underline{M}$ for cyclic $p$-groups |
| topic | Algebraic Topology 55Q91 (Primary), 55P91 (Secondary) |
| url | https://arxiv.org/abs/2309.17242 |