Stable moduli interpretation of $\mathbb{R}\mathrm{P}^\infty_{-1}$

Question

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.)

Answer 1

Let $T$ denote the tautological bundle over $BO(d)$ or $BSO(d)$. Suppose you have a bundle $E\to B$ of smooth closed $d$-manifolds, with vertical tangent bundle $V$. Using a Gysin map and a classifying map we get maps $\Sigma^\infty B_+\to E^{-V}\to BO(d)^{-T}$ of Thom spectra, and thus a map $B\to\Omega^\infty BO(d)^{-T}$ of spaces. With orientation conditions we can change the target to $\Omega^\infty BSO(d)^{-T}$. When $d=2$ we have $SO(2)=S^1$ and so the target is $\Omega^\infty \mathbb{C}P^\infty_{-1}$. Madsen and Weiss proved that when $E\to B$ is the universal bundle of Riemann surfaces of high genus, the map $B\to\Omega^\infty\mathbb{C}P^\infty_{-1}$ is homologically highly connected (but the fundamental groups are different unless you do some kind of plus construction or group completion to the source). There are various generalisations due to various combinations of Madsen, Weiss, Galatius and Randall-Williams, I don't have a good knowledge of the precise state of play. Anyway, for any not-necessarily-orientable bundle $E\to B$ of smooth closed 1-manifolds you get a map $B\to\Omega^\infty\mathbb{R}P^\infty_{-1}$. I don't know if there is a universal example where this map is an equivalence.