TITLE: complex cobordism from formal group laws?
QUESTION [14 upvotes]: Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. Perhaps the corresponding infinite loop space is the classifying space for some category defined in terms of formal group laws. Infinite loop space theorists, where are you?". What is the state of things on that now?

REPLY [3 votes]: There is a very easy theorem along much weaker but related lines in Adams's "Stable Homotopy and Generalized Homology."  I refer to Lemma 4.6 of section II.
It isn't written quite like this, but essentially it says that if E is a complex orientable spectrum together with a complex orientation $x \in E^{2}(\mathbf{C}P^{\infty})$ then there is a unique (up to homotopy) map of ring spectra 
$MU \rightarrow E$ taking the fixed (better fix one) complex orientation of $MU$ to the given complex orientation of $E$.
This doesn't build $MU$ out of formal group laws of course, but it shows $MU$ has this universal property for complex oriented cohomology theories, and this book was around of course when Ravenel wrote the green book.
It does seem like with the modern point of view these ideas should yield a construction out of complex oriented theories if not quite out of formal group laws.