The Brown-Comenetz dual of the K(2)-local sphere at the prime 3

TL;DR

This paper computes the homotopy type of the Brown-Comenetz dual of the K(2)-local sphere at prime 3, revealing a non-trivial twist by an element in the Picard group and providing a detailed characterization.

Contribution

It provides the first explicit calculation of the Brown-Comenetz dual at this chromatic level and prime, including the identification of a non-trivial Picard twist.

Findings

Homotopy type of the Brown-Comenetz dual $I_2$ at prime 3 is determined.

A non-trivial element $P$ in the Picard group causes a twist in the dual.

Complete characterization of the twisting element $P$ is achieved.

Abstract

We calculate the homotopy type of the Brown-Comenetz dual $I_2$ of the K(2)-local sphere at the prime 3 and show that there is a twisting by a non-trivial element $P$ in the exotic part of the Picard group. We give a complete characterization of $P$ as well. The main technique is to give a sequence of calculations of the homotopy groups of elements of the Picard group after smashing with the Smith-Toda complex V(1).