It is a well known fact that (isomorphism classes of) *principal* $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$.

There is always the trivial one $B\times S^1$, and for $B=S^3$ for example, these are all. The Hopf bundle is an interesting example when $B=S^2$. The principal bundles over $S^2$ correspond bijectively to $\mathbb{Z}$; much interesting information about them can be extracted using tools such as in Blair[1].

However, over the Klein bottle or $\mathbb{R}P^2$ there is only one such non-trivial bundle (since their second cohomology is $\mathbb{Z}_2$). So in order to classify all of them, one just needs to find out what is the non-trivial bundle.

# Question:

- What are the
**non-trivial***principal*circle bundles over the Klein bottle and $\mathbb{R}P^2$ ? In particular, given a nice base-space with 2nd integral cohomology a finite group (such as $\mathbb{R}P^2$), is there a constructive way to find out what are the

*non-trivial*principal circle bundles over it?[1]: D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds.

Fiber Bundlesbook as well as many other bundle books and standard obstruction theory references. $\endgroup$