I'm reading Chapter 10.4 on the 't Hooft-Polyakov monopoles in Ryder's Quantum Field Theory.

On page 412 he explains why magnetic monopoles cannot appear in the Weinberg-Salam model.
I'm I right by saying that he shows that the electromagnetic gauge group $U(1)_{em}$ is not compactly embedded into the $U(1)\times U(1)$ subgroup of $SU(2)\times U(1)$?

He then immediately concludes that the first fundamental group of the unbroken symmetry, which is $H=U(1)_{em}$, $\pi_1(H)$ must be trivial or doesn't exists. Could someone refer me why?

Comment: I know that in the $SU(2)\times U(1)$ ones must consider the second homotopy group from $S^2$ to the orbit $G/H=SU(2)\times U(1)/U(1)$, where $H$ is the isotropy group of a vacuum state, after symmetry breaking. But the second homotopy group of a quotient can be related through a exact series to the kernel of the map from $\pi_1(H)$ into $\pi_1(G)$.

What I do not understand is by which theorem for $H$ having a non-compact covering group $\pi_1(H)$ must be trivial or non-existing (???)?

This post imported from StackExchange Physics at 2014-05-04 11:10 (UCT), posted by SE-user Anne O'Nyme