Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups:

• $(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth homotopy equivalence, and $(g\cdot h)(p):=g(p)\cdot h(p)\in G$.
• $(C^\infty(M;G)/{\sim},\circ )$: Here, $\circ$ is composition as in $\pi_3(G)$.

Based on Witten's paper QFT and the Jones Polynomial, these smooth homotopy groups must be isomorphic. If the maps in the respective homotopy groups are compactly supported, I can see separating the supports of two maps via homotopy to induce a group homomorphism.

However, all this reasoning seems vague. Does anyone know of the details for these arguments or references where I may read similar arguments fleshed out (for example, how to write a homotopy which separates two maps in a non-euclidean setting)?