Adams spectral sequence of tmfRP{\mathrm{tmf}}\wedge {\mathbb {RP}}^\infty Stems 97–192

5 Stems 97–192

The situation in stems 97–192 is very similar to the first 96 stems. In the Adams spectral sequence for tmf{\mathrm{tmf}}, we have

d3(v216)=w19. d_3(v_2^{16}) = w^{19}.

So in particular, all d2d_2's in this range are the same as in the first 96. Moreover, by the Leibniz rule, for any xx, we have

d3(xv216)=d3(x)v216+w19x. d_3(x v_2^{16}) = d_3(x) v_2^{16} + w^{19} x.

For most of the terms, the ww multiples have already been killed by d2d_2's, so the d3d_3's get preserved. The extra d3d_3's we get come from ww multiples of v216x1,0v_2^{16} x_{1, 0}, v224x1,0v_2^{24} x_{1,0} and v216x20,3v_2^{16} x_{20, 3}.

The d4d_4's are also preserved by v216v_2^{16}, except for the d4d_4 on w4x35,6w^4 x_{35, 6}, which supports a d3d_3 instead. This follows from the fact that our d4d_4's are v14v_1^4 periodic and v14v216v_1^4 v_2^{16} is permanent.

We depict the interesting d3d_3's in Figure 8, omitting the classes that get killed by differentials propagated from the first 96 stems. This is included because one has to do a bit of book keeping to keep track of which of the w19+kw^{19 + k} multiples actually get killed for small kk.

\begin{sseqpage} [name = tmf ass later, page = 3]
    \draw[differential style](117, 20, 1) -- (116, 23);
Figure 8 Differentials in 96–192
\begin{sseqpage} [name = tmf ass later, page = 4]
    \structline[hone](117, 20)(118, 21)
Figure 9 Permanent classes in 96–192