The Tate spectrum of the higher real $K$-theories at height $n=p-1$

TL;DR

This paper computes the Tate spectrum of higher real $K$-theories at height $n=p-1$, showing it is contractible for maximal finite $p$-subgroups at odd primes, and relates it to homotopy fixed points and orbits.

Contribution

It provides the first calculation of the Tate spectrum at height $n=p-1$ for maximal finite $p$-subgroups, establishing its contractibility and connecting it to known homotopy fixed point spectra.

Findings

$ ext{pi}_*E_{p-1}^{tG} ext{ is trivial}$ for maximal finite $p$-subgroups

The norm map induces an equivalence between homotopy fixed points and orbits

Calculation of $ ext{pi}_*E_{p-1}^{hG}$ matches folklore results

Abstract

Let $E_n$ be Morava $E$-theory and let $G \subset G_n$ be a finite subgroup of $G_n$, the extended Morava stabilizer group. Let $E_{n}^{tG}$ be the Tate spectrum, defined as the cofiber of the norm map $N:(E_n)_{hG} \to E_n^{hG}$. We use the Tate spectral sequence to calculate $\pi_*E_{p-1}^{tG}$ for $G$ a maximal finite $p$-subgroup, and $p$ an odd prime. We show that $E_{p-1}^{tG} \simeq \ast$, so that the norm map gives equivalence between homotopy fixed point and homotopy orbit spectra. Our methods also give a calculation of $\pi_*E_{p-1}^{hG}$, which is a folklore calculation of Hopkins and Miller.