For a closed manifold $X$, suppose $b_2(X)=0$.

Question: For any complex line bundle $L\to X$, can we always find a flat connection, i.e. $\exists$ a connection $\nabla$, such that $\nabla^2=0$.

For a closed manifold $X$, suppose $b_2(X)=0$.

Question: For any complex line bundle $L\to X$, can we always find a flat connection, i.e. does there exist a connection $\nabla$, such that $\nabla^2=0$?