Nontrivial U(1) bundles of a 3-manifold

2018-07-22 02:49:30

If G is a Lie group which is (1) a connected,

(2) simply connected

(3) compact,

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:

$S^3$

$\mathbb{T}^3$

$S^2 \times S^1$

$D^2 \times S^1$

$D^3$

($D^d$ 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?