# The Omega spectrum for mod 2 KO-theory

## TL;DR

This paper explores the structure of the Omega spectrum related to mod 2 KO-theory, computing homology Hopf algebras, describing numerous maps and spectral sequences, and clarifying relationships among various K-theories.

## Contribution

It provides a detailed computation of the homology Hopf algebras and describes all maps and spectral sequences among the involved spectra, including new spaces.

## Key findings

- Computed homology Hopf algebras for the Omega spectrum spaces.
- Described all 98 maps and spectral sequences between the spectra.
- Documented the maps on homotopy for the involved spectra.

## Abstract

The 8-periodic theory that comes from the KO-theory of the mod 2 Moore space is the same as the real first Morava K-theory obtained from the homotopy fixed points of the Z/(2) action on the first Morava K-theory. The first Morava K-theory, K(1), is just mod 2 KU-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum. There are a lot of maps into and out of these spaces and the spaces for KO- theory, KU-theory and the first Morava K-theory. For every one of these 98 maps (counting suspensions) there is a spectral sequence. We describe all 98 maps and spectral sequences. 48 of these maps involve our new spaces and 56 of the spectral sequences do. In addition, the maps on homotopy are all written down.