TITLE: Witten's QFT and Jones Poly paper
QUESTION [10 upvotes]: Data: $M$ is an oriented 3-dim manifold, $E$ is a $G$-bundle over $M$, with $G$ compact 
simple Lie group. 
Question: How does $\pi_3(G)\cong \mathbb{Z}$ imply that there exists non-trivial 
gauge transformations (i.e., continuous maps $M\rightarrow G$ which are not homotopic to 
the trivial map)?
If anyone would like to read it from the source, check the paragraph leading up to 
equation 1.4.

REPLY [4 votes]: @kwl1026. Gauge transformations are sections of the Ad bundle $P\times_{Ad} g$ where $P\to M$ is the principal $G$ bundle; $g$ the lie algebra. When $G$ is abelian the adjoint action is trivial so, e.g.  the $U(1)$ gauge group is always $Map(M,U(1))$ whether or not $P$ is trivial. Its homotopy classes are then $[M, U(1)]= H^1(M;Z)$, which is zero (for $M$ a closed 3-manifold) if and only if $M$ is a rational homology sphere.
An elementary answer to your original question for $SU(2)=S^3$ is that  obstruction theory shows that the primary obstruction gives an isomorphism  $[M,S^3]\to H^3(M;Z)$. An induction using the fibration $SU(n)\to SU(n+1)\to S^{2n+1}$ and cellular approximation shows that $[M,SU(n)]=[M,SU(2)]$. Other tricks can get you there for other $G$. It is true that the differnence in Chern-Simons invariants (suitably normalized) coincides with this isomorphism (composed with $H^3(M;Z)\to Z$), as indcated by Konrad. For $SU(2)$ it also agrees with the degree, as mentioned by Peter.
If $P$ is non-trivial you have to work a little harder, since you are asking what is the set of homotopy classes of sections of the fiber bundle  $P\times_{Ad} g$. A useful reference is Donaldson's book on Floer homology.