# Formal group law of unoriented cobordism

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.

Is there a similar description of $F_O(X,Y)$?

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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.)
Hi Neil. You were right, of course. As I've just got round to checking, the proof on the page I linked goes through $\mathbb{R}$ replacing $\mathbb{C}$. –  Mark Grant Feb 10 '12 at 21:17