The $RO(C_{2^n})$-graded homotopy of $H\underline{\mathbb{Z}}$ through generalized Tate squares

TL;DR

This paper introduces a novel method using generalized Tate squares to compute the $RO(C_{2^n})$-graded homotopy groups of $Hrac{rac{7Z}}{7f}$, providing explicit calculations for the $C_4$ case and spectral sequence analysis.

Contribution

It develops a new approach employing generalized Tate squares to determine equivariant homotopy groups as Green functors, with explicit computations for specific cases.

Findings

Complete computation of $C_4$ case homotopy groups.

Analysis of two $rac{7P}$-homotopy limit spectral sequences.

Demonstration of the method's effectiveness for equivariant spectra.

Abstract

We propose a new method to compute the $C_{2^n}$-equivariant homotopy groups of the Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ as a $RO(C_{2^n})$-graded Green functor using the generalized Tate squares. As an example, we completely compute the $C_4$ case and investigate two $\mathscr{P}$-homotopy limit spectral sequences for the family $\mathscr{P}=\{e,C_2\}$.