It is well known that the formal group law $F\_U$ of complex cobordism, expressing the Euler class of a tensor product of complex line bundles, is universal.


Also, the formal group law $F\_O$ of unoriented cobordism, expressing the Euler class of a tensor product of real line bundles, is universal among formal group laws in characteristic 2 with the property that $F(X,X)=0$.


There is a nice description of $F\_U$ in terms of manifold generators, due to Buchstaber:
$$
F\_U(X,Y) = \frac{\sum\_{i,j\geq 0} [H\_{ij}]\;X^iY^j}{\left(\sum\_{r\geq 0}[\mathbb{C}P^r] X^r\right) \left(\sum\_{s\geq 0}[\mathbb{C}P^s] Y^s\right)}
$$
where the $H\_{ij}$ are Milnor hypersurfaces. Here I am quoting [this page](http://www.map.him.uni-bonn.de/Formal_group_laws_and_genera#Formal_group_law_of_geometric_cobordisms).


Is there a similar description of $F\_O(X,Y)$?



Formal group law of unoriented cobordism
I'm fairly sure you just get the same formula, with $\mathbb{C}P^k$ replaced by $\mathbb{R}P^k$, and $H\_{ij}$ replaced by the corresponding real hypersurface in $\mathbb{R}P^i\times\mathbb{R}P^j$. The proof of the equivalent formula
$$ \left(\sum [\mathbb{R}P^r]\;X^r\right)
 \left(\sum [\mathbb{R}P^s]\;Y^s\right) 
 F\_O(X,Y) = 
 \sum H\_{ij} X^i Y^j
$$
is quite direct and geometric. (I might come back and write more tomorrow.)


