A Thom spectrum from "doubled" tautological bundles?

Question

Let us consider real vector bundles, and denote by $V_k$ the tautological bundle $V_k\to BO(k)$. From $$ Thom(V\oplus 1_{\mathbb{R}}\to X)=\Sigma Thom(V\to X) $$ and from $j^*V_{k+1}=V_k\oplus1_{\mathbb{R}}$, where $j\colon BO(k) \to BO(k+1)$ is the canonical embedding, one sees that writing $MO_k=\Sigma^{-k}Thom(V_k\to BO(k))$ one gets a sequence of morphisms $MO_k\to MO_{k+1}$ and one can define $$ MO=\lim_{\to} MO_k $$ This way (and taking infinite suspension) one defines the Thom spectrum $MO$, whcih can therefore informally be thought as the infinite desuspension of the Thom spectrum of the tautological infinite rank vector bundle over $BO(\infty)$. Direct sum of vector bundles and the fact that $j_{m,n}^*V_{m+n}=V_m\oplus V_n$, where $j_{m,n}\colon BO(m)\times BO(n)\to BO(m+n)$ is the canonical embedding makes $MO$ a ring spectrum.

Apparently, one should be able to do the very same construction by "doubling" the $V_k$'s, i.e., by considering vector bundles $$ W_k=V_k\oplus V_k $$ noticing that $j^*W_{k+1}=W_k\oplus 1_{\mathbb{R}^2}$ and so $Thom(j^*W_{k+1}\to BO(k)) = \Sigma^2 Thom(W_{k}\to BO(k))$, and then considering the pointed spaces $M_2O_k=\Sigma^{-2k}Thom(W_k\to BO(k))$ and then the colimit $$ M_2O=\lim_{\to} M_2O_k $$ This again should be a ring spectrum by the same reason as for $MO$.

As, for any real vector space $V$, its double $V\oplus V$ carries a canonical comple structure $J\colon V\oplus V \to V\oplus V$, and since the defining representation $\mathbb{C}^n$ of $U(n)$ restricted to $O(n)$ splits as the sum of two copies of the defining representation $\mathbb{R}^n$ of $O(n)$, one should get a morphism of ring spectra $M_2O \to MU$.

By looking at how the $\hat{A}$ polynomial is derived from the Todd polynomial, one would suspect that the analogous morphism $$ M_2Spin \to MU $$ and a compatibility between this, the standard complex orientation of $KU$, the Atiyah-Bott-Shapiro orientation of $KO$ and complexification of vector bundles is what secretely lies behind the Atiyah-Singer formula for the index of a twisted spin complex, making Atiyah-Singer formula a version of the general kind of Hirzebruch-Riemann-Roch formulas one has when dealing with morphisms of generalized cohomology theories and pushforwards (see, e.g., Panin-Smirnov).

However, I've not been able so far to locate something resembling $M_2O$ or $M_2Spin$ in the literature I have searched in, so I could possibly be on a false track here.

Is the construction sketched above possibly correct? What is a reference to it? has it a more canonical name than $M_2O$? is it really related to the construction and relevance of $\hat{A}$ along the lines sketched above?

Answer 1

I think that both of your examples, $M_2Spin$ and $M_2O$, arise naturally in the context of Thom spectra induced by $(B,f)$-structures. Given a $(B,f)$-structure $\mathcal{B}= \{f_n: B_n \to BO(n)\}$, the associated Thom spectrum $M\mathcal{B}$ is defined componentwise as: $$ M\mathcal{B}_k = Thom(f_k^*V_k\to B_k), $$ where the maps $\Sigma M\mathcal{B}_k \to M\mathcal{B}_{k+1}$ are given by looking at the pullback square $\require{AMScd}$ $$ \begin{CD} \mathbb{R}\oplus f_k^*V_k @>>> f_{k+1}^*V_{k+1}\\ @VVV @VVV \\ B_k @>>> B_{k+1}. \end{CD} $$ One can 'double' this construction replacing the maps $f_k$ with $\tilde{f_k}$ defined to be the composition: $$ \tilde{B_{2k}}:=B_k\overset{f_k}{\to} BO(k) \overset{\Delta}{\to} BO(k) \times BO(k) \overset{j_{k,k}}{\to} BO(2k) $$ and get a $S^2$-$(B,f)$-structure, a $(B,f)$-structure indexed only on even natural numbers, denoted by $2\mathcal{B}$. By definition, the Thom spectrum $M_2\mathcal{B}$ associated to this new $S^2$-$(B,f)$-structure, is

$$ (M_2\mathcal{B})_{2k} = Thom(\tilde{f_k^*}V_{2k}\to \tilde{B_{2k}}) = Thom(f_k^*V_k\oplus f_k^*V_k \to B_k) $$ $$ (M_2\mathcal{B})_{2k+1} = \Sigma(M_2\mathcal{B})_{2k}. $$ In your case, $M_2Spin$ and $M_2O$, are (as sequential spectra) the Thom spectra associated to the 'doubled' $(B,f)$-structures that classically define $MSpin$ and $MO$, i.e. the $(B,f)$-structures respectively given by the maps $BSpin(k)\to BO(k)$ and ${{\rm id}}_{BO(k)}$.

All the details and references can be found in the nlab pages Thom spectrum and G-structure.