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We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion $\mathbb{C}P^1\hookrightarrow \mathbb{C}P^\infty$. Is there a connection between this notion of "complex orientation" and the notion of a Thom spectrum being $R$ oriented with respect to a ring spectrum $R$ in the sense of ABGHR? That is, for a Thom spectrum $Mf$ associated to a map $f:X\to BGL_1\mathbb{S}$, and a ring spectrum $R$, we say that $Mf$ is $R$-oriented if the composition $X\to BGL_1\mathbb{S}\to BGL_1R$ is null. If we consider $MU$ to be the Thom spectrum associated to some map (I'm not sure which it should be, but I suspect this is the way it's done) $BU\to BGL_1\mathbb{S}$, can we rephrase the notion of complex orientation in this language?

Thanks!

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What is ABGHR? I guess you already know that a complex orientation on $E$ is the same as a map of ring spectra $MU\to E$. –  Mark Grant Feb 12 '13 at 19:57
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Sorry, ABGHR is the paper: arxiv.org/abs/0810.4535 –  Jon Beardsley Feb 12 '13 at 20:49
    
And yeah, thanks Mark. I'm trying to figure out how that's related. –  Jon Beardsley Feb 12 '13 at 20:51
    
This doesn't help answer your real question but gives your map f. F= GL_1S is the subspace of QS^0 = colim \Omega^n\Sigma^n consisting of stable maps of spheres of degree 1 or -1. Let SF(n) be the monoid under composition of maps S^n\to S^n of degree 1. Using one-point compactification of C^n$, U(n) embeds in SF(2n); passing to colimits gives U \to SF; passing to classifying spaces gives f: BU \to BSF. But the map is irrelevant to the classical construction of the Thom spectrum MU, which just uses the Thom spaces of the universal complex n-plane bundles directly. –  Peter May Feb 13 '13 at 5:14
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up vote 7 down vote accepted

When I wrote the comment above, my memory was blanking. The connection between ring maps $MU\to R$ and complex orientations that Mark describes goes back to Quillen's original work relating $MU$ to formal group laws. (Lemma 4.6, page 52, in Adams Stable Homotopy and Generalized Cohomology characterizes ring maps $MU\to R$ in terms of complex orientations). A discussion of $E_{\infty}$ orientations $MU\to R$ that may help with the original question is given in Section 5 of ``What are $E_{\infty}$ ring spaces good for?'', which relates spectrum level orientations to $E_{\infty}$ orientations on the space level. Such orientations are equivalent to $E_{\infty}$ maps $BU \to B(U;R)$, where $B(U;R)$ classifies $R$-oriented $U$-bundles (in the classical sense of orientation). I apologize if the relevance is unclear. It's late.

[Continued] Here is some more background. To understand the mathematics here, you must recognize that the unit space once called $F$ and now called $GL_1(S)$ plays two very different roles, one additive and one multiplicative. This is explained in the introduction to "What are $E_{\infty}$ ring spaces good for?" The space $BF$ classifies sectioned stable spherical fibrations, and its product classifies fiberwise smash products. You go from stable vector bundles to stable spherical fibrations by fiberwise one point compactification, and that takes Whitney sum of bundles to fiberwise smash product. Therefore we think of this as additive structure. The map $BU \rightarrow BF$ sees this on the represented functor level, and it is maps like this that you are thinking of as leading to Thom spectra (as they do by Gaunce Lewis's thesis, in "Equivariant stable homotopy theory" LMS 1213. There is also an infinite loop space $BU_{\otimes}$, which is the identity component of the space $GL_1(KU)$. There is no analogous $BF_{\otimes}$.

As the unit space $GL_1(S)$, we think of the same space $F$ as multiplicative. For any commutative (ie $E_{\infty}$) ring spectrum $R$, we have the unit map $S\to R$, and on passage to zeroth spaces it induces an infinite loop map $GL_1(S) \rightarrow GL_1(R)$. There is a fibration sequence $B(U,R) \to BU \to BGL_1(R)$. The first map says ``forget the orientation''. The second is the obstruction to (universal) $R$-orientability of complex vector bundles. An $E_{\infty}$ $R$-orientation of complex bundle theory is an $E_{\infty}$ map $g\colon BU \to B(U,R)$ which sections $B(U,R) \to BU$. Such orientations $g$ correspond to maps $MU\to R$ of commutative ring spectra, as I explained in Section 5 opus cit. Of course, everything in this general theory works equally well for other kinds of bundles. The interplay of additive and multiplicative structure is the key to the splitting of $SF$ as $J\times Coker J$ at each prime $p$ (only an $E_{\infty}$ splitting when $p$ is odd).

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