Adams spectral sequence of ${\mathrm{tmf}}\wedge {\mathbb {RP}}^\infty$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 ${\mathrm{tmf}}$, we have

$d_3(v_2^{16}) = w^{19}.$

So in particular, all $d_2$'s in this range are the same as in the first 96. Moreover, by the Leibniz rule, for any $x$, we have

$d_3(x v_2^{16}) = d_3(x) v_2^{16} + w^{19} x.$

For most of the terms, the $w$ multiples have already been killed by $d_2$'s, so the $d_3$'s get preserved. The extra $d_3$'s we get come from $w$ multiples of $v_2^{16} x_{1, 0}$, $v_2^{24} x_{1,0}$ and $v_2^{16} x_{20, 3}$.

The $d_4$'s are also preserved by $v_2^{16}$, except for the $d_4$ on $w^4 x_{35, 6}$, which supports a $d_3$ instead. This follows from the fact that our $d_4$'s are $v_1^4$ periodic and $v_1^4 v_2^{16}$ is permanent.

We depict the interesting $d_3$'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 $w^{19 + k}$ multiples actually get killed for small $k$.