I am currently reading an article about TFTs (DW - Group Cohomology and TFTs), and I have a few questions:

(1) Let $M$ be a 3-man., then we know there exists some 4-man. $B$ such that $\partial B = M$. Now, let $E$ be a $G$-bundle over $M$, when can we extend $E$ to a $G$-bundle over $B$ and a connection $A$ over $E$ to a connection $A'$ over $E'$?

(2) If we can construct such an $E'$ over $B$ then we can re-define the CS action as $CS(A') = \frac{k}{8\pi^2}\int_B Tr(F'\wedge F')$, where $F'$ is the curvature of $A'$. According to the paper, if $k$ is an integer then $CS(A')$ is independent, mod 1, of the choice of $B$ and of the extension $E$ and $A$. How do you see this?

(3) Later on they give a better def of the action, $CS = \frac{1}{n}\left(\int_B \frac{k}{8\pi^2}Tr(F'\wedge F') - \langle \gamma^*\omega,B\rangle\right)$, where $\gamma : B \rightarrow BG$ and $\omega$ is an integer-valued cocycle. They then go on to say that if $B$ is closed then $\int_B \frac{k}{8\pi^2}Tr(F'\wedge F') = \langle \gamma^*\omega,B\rangle$ and so $CS$ is independent of $B$ and the way we have continued the bundle and the connection. Why is this so?