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?



Questions from Chern-Simons theory
(1) Obstruction theory: there is a sequence of obstructions to extending a bundle On $M$ to all of $B$. For homotopy theoretic purposes, a bundle is the same as a map to $BG$. The obstructions live in $H^{n}(B,M;\pi\_{n-1}(BG))$. These groups are trivial for a $4$-manifold $B$ when $BG$ is $3$-connected, which happens as along as $G$ is connected and simply-connected.


(2) Chern-Weil theory: $\frac{1}{4 \pi^2}F \wedge F$ represents the first Pontrjagin class, which is integral. If you have two choices of nullbordism, glue them together to get a closed manifold. The difference (because of orientation) of the integrals over the two nullbordisms is the same as the integral over the closed manifold. The role of the $k$ is not so clear to me.


(3) The cocylce $\omega$ has to be $k$ times the Pontrjagin class, then this is true. The same argument as in (2) should give the result.


