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

3 ww-periodic classes

Next, we look at the orange and yellow classes. The yellow classes is a free k[w]k[w]-module on a single generator x31,6x_{31, 6}, where w=(5,1)|w| = (5, 1). There is no actual element ww that you can multiply with; rather, it is given by the Massey product h1,h2,\langle h_1, h_2, -\rangle .

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

β=w3g=w4γ=w5 \begin{aligned} \beta & = w^3\\ g & = w^4\\ \gamma & = w^5 \end{aligned}

So wkw^k exists for k3k \geq 3 and is permanent for sufficiently large kk. These yellow classes are in fact all permanent.

It is useful to note that x31,6x_{31, 6} is in fact itself ww divisible, with w6x1,0=x31,6w^6 x_{1, 0} = x_{31, 6}. Note that w4x1,0=gx1,0w^4 x_{1, 0} = g x_{1, 0} is the sum of the two basis elements in bidegree (21,4)(21, 4). The two basis elements can be uniquely identified as follows — the brown class is the unique non-zero class that is v14v_1^4 torsion, while the black class is the unique non-zero class that is h1h_1 divisible.

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

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

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

The orange classes form a free k[w,v1]k[w, v_1]-module on a single generator x35,6x_{35, 6}. Again v14v_1^4 exists and is given by, well, v14v_1^4. It is also convenient to note that

d0=w2v12e0=w3v1α=w2v1 \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 tmf{\mathrm{tmf}} and the Leibniz rule. It is prudent to note that the Adams spectral sequence for tmf{\mathrm{tmf}} has

d2(v28)=gαβ=v1w9, d_2(v_2^8) = g \alpha \beta = v_1 w^9,

so w9+kv11+jx35,6=0w^{9 + k} v_1^{1 + j} x_{35, 6} = 0 for all j,k0j, k \geq 0 on the E3E_3 page, and we are left with a sequence w9+kx35,6w^{9 + k} x_{35, 6} of dots separated by (5,1)(5, 1). In fact the sequence starts at x20,3x_{20, 3} with w2x20,3=x30,5w^2 x_{20, 3} = x_{30, 5} and w3x20,3=x35,6w^3 x_{20, 3} = x_{35, 6}. These again eventually get killed by d3d_3's in a manner exactly analogous to the x31,6x_{31, 6}'s.