C_pⁿ-equivariant Mahowald invariants

2024/11/01 by William Balderrama, Yueshi Hou, Balderrama, William +3
Mathematics · #Mathematics and Applications #Geometric Analysis and Curvature Flows #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2411.00421

Abstract

The classical Mahowald invariant is an operation that systematically produces new elements in the stable homotopy groups of spheres from known ones. We introduce the C_pⁿ-Mahowald invariant: a relation π_⋆ S_C_p^n-1 \rightharpoonup π_∗ S between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when n=1. We compute the C_pⁿ-Mahowald invariants of all elements in the Burnside ring A(C_p^n-1) = π₀ S_C_p^n-1, extending Mahowald and Ravenel's computation of M_Cp(p^k). As a consequence, we determine the image of the C_p-geometric fixed point map Φ^C_p : π_V S_C_pⁿ → π₀ S_C_pⁿ/C_p ≅ A(C_p^n-1) when V is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for n=1.

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