3-connection on nontrivial 3-manifold

Question

I'm studying Chern-Simons theory on topological nontrivial 3-manifold (I come from a physics background, so I'm new to some mathematical concepts). If the first homology group $H_1(M)$ is nontrivial one needs to consider a good cover of the manifold and a polyhedral decomposition. Then, we can define a U(1) gauge field as follows: 3-connection construction.

1. Why if the homology group is trivial i can take $d\lambda_{ab}$ so that $v_a=v_b$, without speaking of bundles?

On the other hand, in the language of bundles, if the manifold is globally trivializable (so admits a global section) the connection could be described by 1-form globally defined.

2. Why if the first homology group is trivial then the manifold is globally trivializable?