TITLE: Why is $\pi_* MU$ concentrated in even degrees?
QUESTION [8 upvotes]: $\pi_* MU$, which is the cobordism ring of manifolds with a complex structure on the stable normal bundle, is a polynomial ring $\mathbb{Z}[x_2, x_4, \dots]$. I'm probably being silly here, but is there some obvious reason why everything is in even degrees? Is it obvious that an odd-dimensional manifold whose normal bundle in some imbedding gets a complex structure is cobordant to zero? (For that matter, is it even obvious that such a manifold is unorientedly cobordant to zero?)

REPLY [4 votes]: According to Peter May's Stable Algebraic Topology 1945-1966 
...there is no known geometric reason why the 
complex cobordism ring should be concentrated in even degree

(This is on page 23)
So really, as others have alluded to, I guess this is just an algebraic result. It would be interesting to find some geometric meaning behind it!