The goal of this note is to outline the computation of the Adams spectral sequence of tmfRP{\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 E2E_2 page, which was done by Davis and Mahowald [1] (in their notation, ΣRP=P1\Sigma ^\infty {\mathbb {RP}}^\infty = P_1). As usual, we have

ExtAs,t(k,H(tmfΣRP))=ExtA(2)s,t(k,H(ΣRP)). \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 v28v_2^8, where v28=(48,8)|v_2^8| = (48, 8). Thus, to understand this group, it suffices to describe the generators under v28v_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 9696 are depicted in ??. The range 9696192192 is fairly similar and is depicted in Figure 8. Finally, v232v_2^{32} is permanent and so all differentials are v232v_2^{32}-periodic.

\begin{sseqpage} [name = tmf ass, page = 1, no virtualclass, x range={1}{60}, y range={0}{32}, scale=1.7]
    \foreach \m in {1,...,7} {
    \cigroup[gray](48, 8){1}

Figure 1 ExtA(2)(k,H(P1))\operatorname{Ext}_{A(2)}(k, H_*(P_1))