I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism. 



> 
> **Do you know any suitable reference for the basics of complex cobordism?**
> 
> 
> 


If possible, I would like the reference to cover a particular result. Let $E$ be a spectrum representing a cohomology theory and $MU$ the spectrum representing complex cobordism. As an analogy with the algebraic case, it should hold that
$$
E^\*(MU)\simeq E^\*(pt)[[c\_1,c\_2, \ldots]]
$$
where $c\_i$ are the universal Chern classes. In other words, $E^\*(MU)\simeq E^\*(Gr)$ where $Gr$ denotes the infinite Grassmanian. If possible, I would like the reference to cover such result and its surroundings.



Reference on complex cobordism
This is worked out in part 2 of



> 
> *Adams, J. F.*, Stable homotopy and generalised homology, Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. X, 373 p. £ 3.00 (1974). [ZBL0309.55016](https://zbmath.org/?q=an:0309.55016).
> 
> 
> 


(note that to understand part 2 you need to have read part 3 first. Yeah, I know)


In particular the result you want is true only for *complex orientable* cohomology theories (those theories for which we can define $c\_1$). This is to be expected, since what you want is essentially a version of the Thom isomorphism, that should hold only if the vector bundle is orientable for your cohomology theory.


Also, Adams works on homology rather than cohomology (it is a lot more convenient when having non-finite spaces like $BU\_n$), but of course once you've proven you have a Thom class, the Thom isomorphism theorem holds in cohomology as well, thus proving the statement you're after.


