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Nontrivial U(1) bundles of a 3-manifold
If G is a Lie group which is (1) a connected,
(2) simply connected
then a G bundle on a 3-manifold is necessarily trivial.
However, U(1) bundle does not satisfy this (2) simply connected criterion.
Can we construct explicit nontrivial U(1) bundles of a 3-manifold? For the following examples:
$S^2 \times S^1$
$D^2 \times S^1$
($D^d$ is a $d$-disk.)
It looks that it is easier to do on $S^2 \times S^1$ if we consider a nontrivial Chern number $c_1$ over the $S^2$ (?). How about other cases?