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I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It seems that later in Hopkins notes he says that the complex orientations of E are in one to one correspondence with multiplicative maps $MU \rightarrow E$, is there a treatment that starts with this perspective? How do the complex orientations of a spectrum E help one compute the homotopy of $E$, or the $E$-(co)homology of MU? Further, what other kinds of orientations could we think about, are there interesting $ko$ or $KO$ orienations? how much of these $E$-orientations of X is detected by E-cohomology of X?

I do have some of the key references already in my library, for example the notes of Hopkins from '99, Rezk's 512 notes, Ravenel, and Lurie's recent course notes. If there are other references that would be great. I am secretly hoping to get some insight from some of the experts. (I guess I should really also go through Tyler's abelian varieties paper)

(sorry for the on and off texing but the preview is giving me weird feedback.)

EDIT: I eventually found the type of answer i was looking for in some notes of Mark Behrens on a course he taught. This answer is that a ring spectrum $R$ is complex orientable is there is a map of ring spectra $MU \to R$. This also appears in COCTALOS by Hopkins but neither source takes this as the more fundamental concept. Anyway, the below answer is more interesting geometrically.

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I sympathize with you, Sean. For a few minutes LaTeX was not rendered for me, even after reloading the page several times. Finally now it is working. –  Georges Elencwajg Mar 3 '10 at 21:47
    
There's Yuli Rudyak's book on "Thom Spectra, Orientations and Cobordism" as well. –  Ryan Budney Mar 23 '10 at 5:43
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up vote 15 down vote accepted

The natural starting point of this story are E-orientations on, say closed, manifolds M. That's just a fundamental class $[M^n] \in E_n(M)$ such that cap product induces a (Poincare duality) isomorphism. Given E, the question becomes which M are E-orientable. In many cases it happens that this follows if the stable normal bundle of M admits a lift through a fibration $X\to BO$. For example, if E=HZ is ordinary Z-cohomology then X=BSO works, if E=KO then X=BSpin works, if E=KU then X=BU or X=BSpin$^c$ works etc.

To formalize the idea that every X-manifold has an E-orientation, form the bordism groups $\Omega^X_n$ of X-manifolds and observe that the fundamental classes lead to natural maps $\Omega^X_n(Y) \to E_n(Y)$ for any space Y. In other words, there are natural transformations of cohomology theories $\Omega^X \to E$, or even better, maps of spectra $MX \to E$, where $MX$ is the Thom spectrum associated to the fibration $X\to BO$.

In the case X=BU this is called a complex orientation of E and has been studied extensively because it simplifies computations of E-cohomology tremendously. The original and still relevant reference is Adams' little blue book.

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