1 ${\mathrm{ko}}$ type classes

We first deal with the gray classes that look like quotients of ${\mathrm{ko}}$s. To understand these classes, we use the cofiber sequence

$${\mathrm{tmf}}^{hC_2} \to {\mathrm{tmf}}^{t C_2} \to \Sigma {\mathrm{tmf}}_{hC_2}$$

which induces a short exact sequence on homology, if we think of ${\mathrm{tmf}}^{hC_2}$ and ${\mathrm{tmf}}^{t C_2}$ as pro-spectra in the usual way. Moreover, by [2, Lemma 1.3], we know that

$$\operatorname{Ext}^{s, t}_c(k, H_*({\mathrm{tmf}}^{t C_2})) \cong \bigoplus _{k \in {\mathbb {Z}}} \operatorname{Ext}_{A(1)}^{s, t}(k, k[8k]).$$

So the $\operatorname{Ext}$ groups of ${\mathrm{tmf}}^{tC_2}$ look like a bunch of ${\mathrm{ko}}$'s, and for degree reasons, its Adams spectral sequence must degenerate.

We claim that the gray classes are in the image of $\operatorname{Ext}^{s, t}_c(k, H_*{\mathrm{tmf}}^{tC_2})$, hence must be permanent. It suffices to prove that the generators under $h_0$ and $v_1^4$ are in the image, i.e. the classes in bidegree $(8k - 1, 0)$. To do so, we note that they cannot be in the image of

$$\operatorname{Ext}^{s, t}_c(k, H_*({\mathrm{tmf}}^{h C_2})) \to \operatorname{Ext}^{s, t}_c(k, H_*({\mathrm{tmf}}^{t C_2})).$$

Indeed, the left-hand side is

$$\varprojlim \operatorname{Ext}^{s, t}_{A(2)} (k, H_*(D \Sigma ^\infty _+{\mathbb {RP}}^n)).$$

The top dimensional cell in $D \Sigma ^\infty _+ {\mathbb {RP}}^n$ is always in degree $0$. So the bigraded group $\operatorname{Ext}^{s, t}_A(k, H_*({\mathrm{tmf}}^{\Sigma ^\infty _+ {\mathbb {RP}}^\infty }))$ has a bottom vanishing line equal to that of $\operatorname{Ext}^{s, t}_{A(2)}(k, k)$. In particular, the corresponding generators at $(8k - 1, 0)$ are all below this line, so are mapped injectively into $\operatorname{Ext}_{A(2)}(k, H_*(P_1))$.

We now give a formal description of these classes. For any $i \in {\mathbb {Z}}$, we let C($i$) be the chart of $\Sigma ^{8i - 1}{\mathrm{ko}}$ truncated to below the line $y = -\frac{x}{4} + 6i - 1$. Then the gray classes are given by $\bigoplus _{i \geq 1}$C($$i$)$$$ plus all its $v_2^8$-multiples. We depict C($3$) in Figure 2 for reference.