Adams spectral sequence of tmfRP{\mathrm{tmf}}\wedge {\mathbb {RP}}^\infty ko{\mathrm{ko}} type classes

1 ko{\mathrm{ko}} type classes

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

tmfhC2tmftC2ΣtmfhC2 {\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 tmfhC2{\mathrm{tmf}}^{hC_2} and tmftC2{\mathrm{tmf}}^{t C_2} as pro-spectra in the usual way. Moreover, by [2, Lemma 1.3], we know that

Extcs,t(k,H(tmftC2))kZExtA(1)s,t(k,k[8k]). \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 Ext\operatorname{Ext} groups of tmftC2{\mathrm{tmf}}^{tC_2} look like a bunch of ko{\mathrm{ko}}'s, and for degree reasons, its Adams spectral sequence must degenerate.

We claim that the gray classes are in the image of Extcs,t(k,HtmftC2)\operatorname{Ext}^{s, t}_c(k, H_*{\mathrm{tmf}}^{tC_2}), hence must be permanent. It suffices to prove that the generators under h0h_0 and v14v_1^4 are in the image, i.e. the classes in bidegree (8k1,0)(8k - 1, 0). To do so, we note that they cannot be in the image of

Extcs,t(k,H(tmfhC2))Extcs,t(k,H(tmftC2)). \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

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

The top dimensional cell in DΣ+RPnD \Sigma ^\infty _+ {\mathbb {RP}}^n is always in degree 00. So the bigraded group ExtAs,t(k,H(tmfΣ+RP))\operatorname{Ext}^{s, t}_A(k, H_*({\mathrm{tmf}}^{\Sigma ^\infty _+ {\mathbb {RP}}^\infty })) has a bottom vanishing line equal to that of ExtA(2)s,t(k,k)\operatorname{Ext}^{s, t}_{A(2)}(k, k). In particular, the corresponding generators at (8k1,0)(8k - 1, 0) are all below this line, so are mapped injectively into ExtA(2)(k,H(P1))\operatorname{Ext}_{A(2)}(k, H_*(P_1)).

We now give a formal description of these classes. For any iZi \in {\mathbb {Z}}, we let C(ii) be the chart of Σ8i1ko\Sigma ^{8i - 1}{\mathrm{ko}} truncated to below the line y=x4+6i1y = -\frac{x}{4} + 6i - 1. Then the gray classes are given by i1\bigoplus _{i \geq 1} C(i)) plus all its v28v_2^8-multiples. We depict C(33) in Figure 2 for reference.

\begin{sseqpage} [scale=0.5, classes = fill, grid = go, grid step = 4]
Figure 2 Depiction of C(33)