Homotopy groups of homotopy fixed point spectra associated to E_n

TL;DR

This paper computes the mod(p) homotopy groups of a specific homotopy fixed point spectrum related to the Lubin-Tate spectrum E_2, exploring implications for duality and finiteness in chromatic homotopy theory.

Contribution

It provides explicit calculations of homotopy groups for E_2^{hH_2} at odd primes and discusses their implications, advancing understanding of homotopy fixed points in chromatic homotopy theory.

Findings

Computed homotopy groups of E_2^{hH_2} for p>2

Analyzed consequences for Brown-Comenetz duality

Discussed finiteness properties of K(n)_*-local spectra

Abstract

We compute the mod(p) homotopy groups of the continuous homotopy fixed point spectrum E_2^{hH_2} for p>2, where E_n is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space of lifts of the height n Honda formal group law over F_{p^n}, and H_n is the subgroup WF^x_{p^n} wreath product Gal(F_{p^n}/F_p) of the extended Morava stabilizer group G_n. We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K(n)_*-local spectra. We also indicate a plan for computing pi_*(E_n^{hH_n} smash V(n-2)), where V(n-2) is an E_{n*}-local Toda complex.