@aaaaaaaaaaaaaaa Would you mind to enclose your math texts within $? It is very difficult to read it (at least for me).

@Dedalus See the above.

@user170039 No problem! Just not sure how the chat room's Latex compiler works, so I haven't been trying to use it

@aaaaaaaaaaaaaaa If you are using a computer/laptop then you may use this extension.

OK thanks!

@skd I don't know sufficient conditions, although I bet someone else does. There is an obvious necessary condition: you need all the elements in $\pi_n(S)$, for $n>0$, to map to zero in your ring spectrum that you want to be complex oriented, since those elements already map to zero in $\pi_*(MU)$. By Joel Cohen's theorem, all such elements are matric Toda brackets of the following elements: $\eta \in \pi_1(S), \nu\in \pi_3(S), \sigma\in \pi_7(S)$...

...and for each odd prime $p$, $\alpha_1\in \pi_{2p-3}(S)$. So if you cone off all those by attaching an $A_{\infty}$-cell (assuming I remember correctly, that an $A_{\infty}$-structure is enough to define Toda brackets of arbitrary length in homotopy?), then you satisfy this necessary condition.

I should say by attaching $A_{\infty}$-cells, not a single $A_{\infty}$-cell, since of course it's going to take more than one (unless you have already localized at an odd prime)

In the case of the free $E_2$-ring with 2=0: isn't $E_2$ just enough to define the Browder bracket? I suppose that means that Toda brackets along with the Browder bracket must be enough to generate all the higher 2-local stable stems from just $2\in \pi_0(S)$, which seems surprising, although maybe I misunderstand something here

@skd Suppose you attach an $E_k$-cell to the sphere spectrum along some element $x\in \pi_*(S)$, and call the result $E_k(x)$. To know that $E_k(x)$ is complex oriented, I think you just need to know that a certain element $w$ in $(E_k(x))^2$ of the $2$-skeleton of $BU(1)$ survives the Atiyah-Hirzebruch SS to give an element in $(E_k(x))^2(BU(1))$. Have you tried working out what the $E_2$-term of that AHSS can look like, in the bidegrees that could be connected to $w$ by a differential?