# Reference request: mod 2 cohomology of periodic KO theory

The mod 2 cohomology of the connective ko spectrum is known to be the module $$\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$$, where $$\mathcal{A}$$ denotes the Steenrod algebra, and $$\mathcal{A}_2$$ denotes the subalgebra generated by $$Sq^1$$ and $$Sq^2$$.

Where can I find the original calculation for referencing it ? What is the structure as module over the steenrod algebra of the mod 2 cohomology of periodic KO theory?

• Unless I'm mistaken $\pi_*(H\mathbb{F}_2\wedge KO)=0$ (it is a ring of positive characteristic containing an isomorphism between the additive and the multiplicative formal group law), so the answer to your second question is rather trivial – Denis Nardin Jul 24 '19 at 16:45
• Indeed, as the periodicity generator ie the Bott element induces zero-map in ordinary homology! – Prasit Jul 24 '19 at 16:56
• It is more usual to use $\mathcal{A}_1$ to denote the algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. – John Palmieri Jul 24 '19 at 19:57

Ravenel in his Complex Cobordism and Stable Homotopy Groups of Spheres attributes this result to Stong, in Determination of $$H^*(BO(k,⋯,∞),Z_2)$$ and $$H^∗(BU(k,⋯,∞),Z_2)$$, but looking at that paper (which is concerned mainly with the "unstable" cohomology of the various constituent spaces of $$ko$$), he attributes this further to
Regarding your second question, the cohomology $$H^*(KO;\mathbb{F}_2)$$ is zero for chromatic reasons (the spectrum $$KU\wedge H\mathbb{F}_2$$ carries an isomorphism of the additive and multiplicative formal group law in characteristic two, so it must be trivial, and then you can use the fact that $$KU=KO/\eta$$ to conclude, since $$\eta$$ is nilpotent in $$\pi_*KO$$), hence the map $$H^*(KO;\mathbb{F}_2)→H^*(ko;\mathbb{F}_2)$$ is trivial.