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. 



Reference on the question mark cell complex
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 $bo\langle i \rangle$.


