I had asked this [question](https://math.stackexchange.com/questions/1712414/classification-of-o2-bundles-in-terms-of-characteristic-classes) in stackexchange but there seems to be no consensus in the answer


It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by their first Chern class. I was wondering what was the characterization of $O(2)$-bundles in terms of characteristic classes. I guess the first and second Setiefel-Whitney classes are necessary for the topological characterization of $O(2)$-bundles, but they can't be enough, because if $w\_{1} = 0$ then one should recover the classification of $SO(2)$-bundles, which is given by the first Chern class and not by the second Stiefel-Whitney class.


Thanks.



Classification of $O(2)$-bundles in terms of characteristic classes
The $O(2)$ bundles $\xi$ over a manifold $M$ are classified by their first Stiefel-Whitney class $w\_1(\xi)\in H^1(M;\mathbb{Z}/2)$ and their twisted Euler class $e(\xi)\in H^2(M;\mathbb{Z}\_{w\_1(\xi)})$.


This is because the space $BO(2)$ is a generalized Eilenberg--Mac Lane space $L\_{w\_1}(\mathbb{Z},2)$ in the sense of 


*Samuel Gitler*, [**Cohomology operations with local coefficients**](http://www.ams.org/mathscinet-getitem?mr=158398), *Amer. J. Math.* **85** (1963), 156--188.


In (slightly) more detail, there is a fibration
$$
K(\mathbb{Z},2)\to E\mathbb{Z}/2\times\_{\mathbb{Z}/2} K(\mathbb{Z},2)\to B\mathbb{Z}/2
$$
given by the twisting of $w\_1$ on the universal $SO(2)$ bundle, and this fibration agrees up to homotopy with the fibration 
$$BSO(2)\to BO(2)\to BO(1).$$ 


