I have read, that the topology and thus the homotopy groups of $[SU(2) \times U(1)] / U(1)$ depend upon the embedding of $U(1)$ into $SU(2) \times U(1)$. For the Electroweak theory inside the standard model, how do I recognise, that the embedding is such, that $\pi_1([SU(2)\times U(1)] / U(1) )= \{e\}$ ? Or how would the Lagrangian explicitly need to look like, such that it has $SU(2)\times U(1)$ gauge symmetry, having broken down Symmetry of $U(1)$, but have a vacuum manifold $M=[SU(2) \times U(1)] / U(1)$, with $\pi_1(M) \neq \{ e\} $ ?
I am referring particularly to the bottom left, last paragraph on the 2nd page of the following paper: https://arxiv.org/abs/1610.05623 (Topology of the Electroweak Vacua)
