I attended a talk recently which closed with the following tantalizing facts: there is a naturally occurring map of spectra $$K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1},$$ which can be rotated to a sequence $$\mathbb{C}\mathrm{P}^\infty_{-1} \to ? \to K(\mathbb{S}).$$ The right-hand spectrum has an interpretation as a sort of stable moduli of finite CW-complexes, and the left-hand spectrum has an interpretation (due to Michael Weiss and someone whose name I've forgotten --- very likely Ib Madsen) as a sort of moduli of Riemann surfaces, stabilized as the genus grows large. The suggestion, then, was that we find some other interesting moduli to fill in the question-mark part of the sequence.

The spectrum $\mathbb{C}\mathrm{P}^\infty_{-1}$ is not so familiar to me. On the other hand, structure encoded in the spectrum $\mathbb{R}\mathrm{P}^\infty_{-1}$ is visible all over homotopy theory (two big e.g.s: the vector fields on spheres problem / Hopf invariant one theorem, and the stabilized EHP spectral sequence / Lin's theorem / the root invariant). Hearing this description of stunted complex projective space has me curious:

Is a stable moduli theoretic description of the homotopy type $\mathbb{R}\mathrm{P}^\infty_{-1}$ known?

(Disclaimer: I'm almost totally ignorant of this area of topology. I'm happy to correct this to a sane question if there's something askew.)