$C_{p^n}$-equivariant Mahowald invariants
William Balderrama, Yueshi Hou, Shangjie Zhang
TL;DR
This work defines and computes the $C_{p^n}$-Mahowald invariant $M_{C_{p^n}}$ that relates equivariant and classical stable stems via the Tate spectral sequence, recovering the classical case when $n=1$. It develops a coherent framework—the Mahowald filtration, spoke grading, and Adams-conjecture reduction—to determine when Burnside-ring elements lift and how fixed-point data control these lifts, yielding explicit descriptions of the invariants for all $X\, ext{in}\,A(C_{p^{n-1}})$. The authors establish precise degree- and torsion-sensitive results, including bases $f_{s,k}^{p,n}$ for degree maps and the $J$-image–torsion interactions that constrain $M_{C_{p^n}}(X)$, with detailed $p$-adic and $2$-primary analyses. By translating the problem to equivariant $K$-theory and Adams operations, they extend classical fixed-point phenomena (Landweber–Iriye) to the $p$-group cyclic setting and connect to richer chromatic equivariant structures. The findings provide a structured, computable description of $igpi_ullet S_{C_{p^n}}$ in representation degrees via the $C_{p^n}$-Mahowald invariant and clarify the image of geometric fixed-point maps in this equivariant context, with explicit consequences for the reduced equivariant stable stems and the $J$-homomorphism.
Abstract
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 $n=1$.