The connective KO-theory of the Eilenberg-MacLane space K(Z_2,2), I: the E_2 page

TL;DR

This paper computes the E_2 page of the Adams spectral sequence for the connective KO-theory of the mod 2 Eilenberg-MacLane space K(Z/2,2), analyzing its cohomology as a module over a subalgebra of the Steenrod algebra.

Contribution

It provides the first step in understanding the KO-theory of K(Z/2,2) by explicitly calculating the E_2 page of its Adams spectral sequence.

Findings

Computed the E_2 page of the Adams spectral sequence for ko_*(K(Z/2,2))

Analyzed the structure of H^*(K(Z/2,2);Z_2) as a module over a subalgebra of the Steenrod algebra

Set the stage for complete spectral sequence analysis in future work

Abstract

We compute the $E_2$ page of the Adams spectral sequence converging to the connective KO-theory of the second mod 2 Eilenberg-MacLane space, $ko_*(K(Z/2,2))$. This required a careful analysis of the structure of $H^*(K(Z/2,2);Z_2)$ as a module over the subalgebra of the Steenrod algebra generated by $Sq^1$ and $Sq^2$. Complete analysis of the spectral sequence will be performed in a subsequent paper.