Adams spectral sequence of ${\mathrm{tmf}}\wedge {\mathbb {RP}}^\infty$$w$-periodic classes

# 3 $w$-periodic classes

Next, we look at the orange and yellow classes. The yellow classes is a free $k[w]$-module on a single generator $x_{31, 6}$, where $|w| = (5, 1)$. There is no actual element $w$ that you can multiply with; rather, it is given by the Massey product $\langle h_1, h_2, -\rangle$.

However, some multiples of $w$ are realized by elements in $\pi _*{\mathrm{tmf}}$, namely

\begin{aligned} \beta & = w^3\\ g & = w^4\\ \gamma & = w^5 \end{aligned}

So $w^k$ exists for $k \geq 3$ and is permanent for sufficiently large $k$. These yellow classes are in fact all permanent.

It is useful to note that $x_{31, 6}$ is in fact itself $w$ divisible, with $w^6 x_{1, 0} = x_{31, 6}$. Note that $w^4 x_{1, 0} = g x_{1, 0}$ is the sum of the two basis elements in bidegree $(21, 4)$. The two basis elements can be uniquely identified as follows — the brown class is the unique non-zero class that is $v_1^4$ torsion, while the black class is the unique non-zero class that is $h_1$ divisible.

These classes stay along for quite a while. The differentials that eventually kill them come from the differential

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

in the Adams spectral sequence for ${\mathrm{tmf}}$.

The orange classes form a free $k[w, v_1]$-module on a single generator $x_{35, 6}$. Again $v_1^4$ exists and is given by, well, $v_1^4$. It is also convenient to note that

\begin{aligned} d_0 & = w^2 v_1^2\\ e_0 & = w^3 v_1\\ \alpha & = w^2 v_1 \end{aligned}

These have a fairly complicated differential pattern, but these all follow from the differentials in the Adams spectral sequence for ${\mathrm{tmf}}$ and the Leibniz rule. It is prudent to note that the Adams spectral sequence for ${\mathrm{tmf}}$ has

$d_2(v_2^8) = g \alpha \beta = v_1 w^9,$

so $w^{9 + k} v_1^{1 + j} x_{35, 6} = 0$ for all $j, k \geq 0$ on the $E_3$ page, and we are left with a sequence $w^{9 + k} x_{35, 6}$ of dots separated by $(5, 1)$. In fact the sequence starts at $x_{20, 3}$ with $w^2 x_{20, 3} = x_{30, 5}$ and $w^3 x_{20, 3} = x_{35, 6}$. These again eventually get killed by $d_3$'s in a manner exactly analogous to the $x_{31, 6}$'s.