In mathematical terms, if you had a G-bundle on spacetime with a connection encoding a gauge field, then if |pi_0(G)| > 1, you could only measure this if pi_1(spacetime) were nontrivial, since the structure group of the bundle would be reducible to the connected component G_0 < G of the identity, if pi_1(spacetime) were trivial. And the connection is valued in the Lie algebra anyway, which only sees G_0.
John Carlos Baez (npub1nf4…nqe4)
