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.)

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    $\begingroup$ Are you certain there's an integral map $K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1}$? I had the impression that one needs to first localize at a prime to get it. There's a map $K(\mathbb{S}) \to TC(\mathbb{S})$ and after localizing at a prime, the target maps to $\mathbb{C}\mathrm{P}^\infty_{-1}$. Am I missing something? $\endgroup$ – John Klein Jan 29 '14 at 22:02
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    $\begingroup$ Oh, no, I'm not sure at all. Not only was I not following the talk perfectly, but as a stable homotopy person I'm susceptible to quietly localizing everything and forgetting about it. $\endgroup$ – Eric Peterson Jan 30 '14 at 2:07

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.

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    $\begingroup$ If this still counts important, by work of Wahl an analogous result of Madsen-Wiess holds in unoriented case: Let $MTO(2)=:BO(2)^{-\gamma_2}$ with $\gamma_2\to BO(2)$ being the canonicla bundle. Then the moduli $\mathcal{N}_\infty$ for unoriented surfaces is homotopy equivalent to $\Omega^\infty MTO(2)$. Now, $\mathbb{R} P_{-1}$ is related to this by a cofibration of spectra $$MTO(2)\to BO(2)_+\to\mathbb{R} P_{-1}$$ which lead to a fibration of infinite loop spaces $$\Omega^\infty MTO(2)\to QBO(2)_+\to \Omega^\infty\mathbb{R}P_{-1}.$$ Found this when searching for `root invariant'. $\endgroup$ – user51223 Aug 22 '15 at 13:57

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