$C_{p^n}$-equivariant Mahowald invariants

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 $C_{p^n}$-Mahowald invariant: a relation $\pi_\star S_{C_{p^{n-1}}} \rightharpoonup \pi_\ast S$ 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 $A(C_{p^{n-1}}) = \pi_0 S_{C_{p^{n-1}}}$, 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 $\Phi^{C_p} : \pi_V S_{C_{p^n}} \to \pi_0 S_{C_{p^n}/C_p} \cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$.