The goal of this note is to outline the computation of the Adams spectral sequence of ${\mathrm{tmf}}\wedge {\mathbb {RP}}^\infty$. Essentially all differentials follow from the Leibniz rule, and products can be computed with a computer. The only work to be done is to organize the computation in order to conclude that we have indeed computed all differentials.

To do so, we need a complete calculation of the Adams $E_2$ page, which was done by Davis and Mahowald [1] (in their notation, $\Sigma ^\infty {\mathbb {RP}}^\infty = P_1$). As usual, we have

$\operatorname{Ext}^{s, t}_{A}(k, H_*({\mathrm{tmf}}\wedge \Sigma ^\infty {\mathbb {RP}}^\infty )) = \operatorname{Ext}^{s, t}_{A(2)} (k, H_*(\Sigma ^\infty {\mathbb {RP}}^\infty )).$

This group is free over $v_2^8$, where $|v_2^8| = (48, 8)$. Thus, to understand this group, it suffices to describe the generators under $v_2^8$. In the Davis–Mahowald description, these generators fall into 4 groups, and we colour-coded these in our chart in Figure 1. We shall go through the different groups in the coming sections, giving a formal description and describe the differentials that pertain to these groups. The differentials up to degree $96$ are depicted in ??. The range $96$$192$ is fairly similar and is depicted in Figure 8. Finally, $v_2^{32}$ is permanent and so all differentials are $v_2^{32}$-periodic.