Reference on the question mark cell complex

The question mark complex is a finite spectrum whose cohomology looks like a "question mark" (when drawn as a module over the Steenrod algebra): that is, there is an element in dimension zero $a_0$, an element $a_2 = \mathrm{Sq}^2 a_0$, and an element $a_6 = \mathrm{Sq}^4 a_2$. It can be constructed by starting with $\Sigma^{-2} \mathbb{CP}^2$, which maps to $S^2$ (thanks to the cofiber sequence $S^1 \to S^0 \to \Sigma^{-2} \mathbb{CP}^2 \to S^2$) and lifting the map $\nu: S^5 \to S^2$ to $\Sigma^{-2} \mathbb{CP}^2$ (which can be done since $\eta \nu = 0$). Then, one takes the cofiber of $S^5 \to \Sigma^{-2}\mathbb{CP}^2$.

Does anyone know any good references on this? I'd like, ideally, to see a few example computations done with it; as it is I don't really have much intuition for how to work with it.

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have you tried constructing a similar complex but with $sq^1$ and $sq^2$ instead? – Sean Tilson Jul 21 '12 at 21:40
Hi Sean. There is such a complex (same logic as above, using $2 \eta = 0$ instead of $\eta \nu = 0$). I'd be interested in seeing computations, e.g., of the stable homotopy of this complex as well. – Akhil Mathew Jul 22 '12 at 1:45
I don't know what the $K$-theory operations look like, actually. The AHSS degenerates and $K^0(?)$ is free on three generators; there's a cofiber sequence $S^0 \to ? \to \Sigma^{-2} \mathbb{HP}^2$ which means that we know the $K$-theory operations except for one: the generator lifting the element of $K^0(S^0)$ could have strange Adams operations --- kind of like what happens in "On the groups J(X) IV." – Akhil Mathew Jul 22 '12 at 12:30
I was thinking you would use an Adams SS to compute its connective real k-theory... – Sean Tilson Jul 24 '12 at 7:57
There are a couple more references, also related to the Picard group - the dual of the question mark complex is referenced in Goerss-Henn-Mahowald-Rezk's "Picard groups at chromatic level 2 for $p = 3$" paper - they at least tell you a bit about its $K$ theory and $KO$ theory. There are also some (hard!) calculations in Ichigi-Shimomura's "$E(2)_*$-invertible spectra smashing with the Smith-Toda spectrum $V(1)$ at the prime 3" (see section 3 in particular) – Drew Heard Dec 19 '12 at 10:36

This is hardly a "canonical example", but one place I've seen the question mark complex $Q$ in action is in Hovey and Sadofsky's paper Invertible spectra in the $E(n)$-local stable homotopy category. There they compute that the even part of the Picard group of the $E(1)$-local stable category at $p = 2$ is generated by (the $E(1)$-localization of) $Q$. As a warning, the exposition at the tail of the paper where this appears is a bit skeletal; you'll be working out the details of the computations and digging up references on your own.
But, in turn, they do cite Hopkins' Minimal atlases of real projective spaces for some facts about $Q$, which I have not read but looks to be useful. He seems to call the complex $N_2$ in section 7, for reasons he explains pertaining to the cohomology of $bo\langle i \rangle$.