# A Thom spectrum from "doubled" tautological bundles?

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?

• For the construction you consider above, the only existing example I can think of are the spectra $\mathbb{R} P^\infty_k$ at the Thom spectrum of $kV_1$ as well as $\mathbb{C} P^\infty_k$ which fit into a nice cofibre sequence of spectra. The other examples that I can think of are Thom spectra of $k\rho_n$ where $\rho_n$ is the reduced representation of $(\mathbb{Z}/2)^{\times n}$. For this, you can look at work of Takayasu. The existing examples somehow show that the spectra $kV_n$ where $k$ is any integer could be/are very complicated. Nov 6, 2019 at 12:27
• @user51223 OP is asking about "Thom spectra over $BO$", and not Thom spectra over $BO(n)$ for fixed $n$. So something that would be more closely related to his question is $MTO$, which would become $M_{-1}O$ in his notation. Nov 6, 2019 at 16:41
• Recall that $MU$ is the Thom spectrum of the bundle over $BU$ given by realification $BU\to BO$. The spectrum $M_2 O$ is the Thom spectrum of the map $BO\to BU\to BO$, where the first map is complexification. Upon Thomification, this gives the desired map $M_2 O\to MU$.
– skd
Nov 9, 2019 at 15:44

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 $${{\rm id}}_{BO(k)}$$.