The $\alpha$-Family in the $K(2)$-Local Sphere at the Prime $2$

TL;DR

This paper computes the image of the alpha-family in the homotopy groups of the $K(2)$-local sphere at prime 2, advancing the understanding of its structure via spectral sequence analysis.

Contribution

It identifies the alpha-family in the $K(2)$-local sphere at prime 2 by locating its image in the algebraic duality spectral sequence, aiding future homotopy computations.

Findings

Determined the image of the alpha-family in the $K(2)$-local sphere at p=2.

Located the alpha-family's image within the algebraic duality spectral sequence.

Facilitated subsequent homotopy group calculations for the $K(2)$-local sphere.

Abstract

In this note, we compute the image of the $\alpha$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.