Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $\pi_*({E_6}^{hC_9})$

TL;DR

This paper advances the understanding of the $p=3$ Kervaire invariant problem by computing the $E_2$-page of the homotopy fixed points spectral sequence for $ ext{pi}_*({E_6}^{hC_9})$, crucial for solving the problem.

Contribution

It provides a detection theorem for the $p=3$ case and computes the $E_2$-page of the spectral sequence using a conjectural $C_9$-action on $E_6$, a novel approach.

Findings

Proved a detection theorem for the $p=3$ Kervaire invariant problem.

Computed the $E_2$-page of the homotopy fixed points spectral sequence.

Used a conjectural $C_9$-action on $E_6$ to facilitate calculations.

Abstract

Hill, Hopkins, and Ravenel suggest that the last remaining Kervaire invariant problem, the case of $p=3$, can be solved by computing the homotopy fixed points spectral sequence for $\pi_* E_6^{hC_9}$. We prove a detection theorem for this case and use a conjectural form of the $C_9$-action on $E_6$ to compute the $E_2$ page of this spectral sequence away from homological degree zero.