Yesterday I read about BFN's construction. One thing that strikes me is the appearance of 'euler class'.

I have never really understood euler class. the input is some bundle over some space, and the output is some cohomology class on the space itself. The only natural thing that I can think of, is the intersection of the zero section with its own perturbation. However, thinking in terms of perturbation is not very intrinsic. this thing should be defined topologically (like what if your underlying space is algebraic, very rigid, no room for deformation? )

then, what do you mean, when you say euler class of a bundle over a point? I would say, the product of weights of that bundle. a weight, is a $\C^*$-valued function on the torus, and it is also a $\C$-valued function on the Lie algebra of the torus. Infinitesimal weights are elements in $t^*$, so products of them lives in $Sym t^*$. That makes sense.

Also, Euler class make sense. The Euler class of a bundle is in degree of the (real rank) of the bundle.

Finally, this is not just ad-hoc definition, it is a calculation using the Borel model. For example, given a standard representation of $G=S^1$, call it $L$, we get a universal bundle $L$ over $BG$. This bundle restricts to $\P^1$ (already visible there) should be given by $u$.

Let's fix the normalization once and for all, and do the BWB computation once in my life. Consider $\C^*$ acting on $\C$. Then, consider the principal $\C^*$ bundle on $\P^1$, so we should get the total space to be $O(-1)$. oops.