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

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 ${\mathrm{\Phi }}^{C_p}:{\pi }_V{S}_{C_{p^n}}\to {\pi }_0{S}_{C_{p^n}\mathrm{/}{C}_p}\cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$.