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

TL;DR

This paper introduces the $C_{p^n}$-Mahowald invariant, extending classical invariants to equivariant stable homotopy theory, computes these invariants for all elements in the Burnside ring, and determines the image of the geometric fixed point map in this context.

Contribution

It defines the $C_{p^n}$-Mahowald invariant, computes it for all elements in the Burnside ring, and extends classical fixed point theorems to the equivariant setting.

Findings

Computed $C_{p^n}$-Mahowald invariants for all elements in the Burnside ring.

Extended classical theorems of Bredon, Landweber, and Iriye to the equivariant case.

Determined the image of the $C_p$-geometric fixed point map for fixed point free representations.

Abstract

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 $n=1$.