Computational trick used in QFT and the Jones Polynomial

Question

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)?

Answer 1

I don't know what you mean by "composition as in $\pi_3(G)$" unless either $M$ is a $3$-sphere or $G$ is, in addition, assumed to be simply connected. If the latter, then $G$ is $2$-connected, and so any map $M \to G$ factors up to homotopy through a map $M \to S^3$ inducing an isomorphism on $H^3$ (here I'm assuming that $M$ is connected for simplicity). So we may assume WLOG that $M = S^3$, and then both groups are isomorphic and isomorphic to $\pi_3(G)$ by the Eckmann-Hilton argument, together with an additional argument about based vs. unbased maps $S^3 \to G$.