$C_{p^n}$-equivariant Mahowald invariants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917961661415424 |
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| author | Balderrama, William Hou, Yueshi Zhang, Shangjie |
| author_facet | Balderrama, William Hou, Yueshi Zhang, Shangjie |
| contents | We introduce the $C_{p^n}$-Mahowald invariant: a relation $π_\star S_{C_{p^{n-1}}} \rightharpoonup π_\ast S$ between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when $n=1$. We compute the $C_{p^n}$-Mahowald invariants of all elements in the Burnside ring $A(C_{p^{n-1}}) = π_0 S_{C_{p^{n-1}}}$, extending Mahowald and Ravenel's computation of $M_{C_p}(p^k)$. As a consequence, we determine the image of the $C_p$-geometric fixed point map $Φ^{C_p} : π_V S_{C_{p^n}} \to π_0 S_{C_{p^n}/C_p} \cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $C_{p^n}$-equivariant Mahowald invariants Balderrama, William Hou, Yueshi Zhang, Shangjie Algebraic Topology We introduce the $C_{p^n}$-Mahowald invariant: a relation $π_\star S_{C_{p^{n-1}}} \rightharpoonup π_\ast S$ between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when $n=1$. We compute the $C_{p^n}$-Mahowald invariants of all elements in the Burnside ring $A(C_{p^{n-1}}) = π_0 S_{C_{p^{n-1}}}$, extending Mahowald and Ravenel's computation of $M_{C_p}(p^k)$. As a consequence, we determine the image of the $C_p$-geometric fixed point map $Φ^{C_p} : π_V S_{C_{p^n}} \to π_0 S_{C_{p^n}/C_p} \cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$. |
| title | $C_{p^n}$-equivariant Mahowald invariants |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2411.00421 |