The Exotic $K(2)$-Local Picard Group at the Prime $2$

TL;DR

This paper computes the structure of the exotic part of the $K(2)$-local Picard group at prime 2, revealing its detailed algebraic composition and introducing new construction techniques involving a $J$-homomorphism.

Contribution

It introduces a novel approach using a $J$-homomorphism from real representations to construct elements in the Picard group at prime 2.

Findings

The exotic $K(2)$-local Picard group at prime 2 has order $2^9$.

The group is isomorphic to $(bZ/8)^2 imes (bZ/2)^3$.

A new technique using a $J$-homomorphism was developed for this calculation.

Abstract

We calculate the group $\kappa_2$ of exotic elements in the $K(2)$-local Picard group at the prime $2$ and find it is a group of order $2^9$ isomorphic to $(\mathbb{Z}/8)^2 \times (\mathbb{Z}/2)^3$. In order to do this we must define and exploit a variety of different ways of constructing elements in the Picard group, and this requires a significant exploration of the theory. The most innovative technique, which so far has worked best at the prime $2$, is the use of a $J$-homomorphism from the group of real representations of finite quotients of the Morava stabilizer group to the $K(n)$-local Picard group.