I understand (i think) that for a magnetic monopole to exist as the result of a gauge group $G$ being spontaneously broken to a subgroup $H$ by the Higgs mechanism, that certain criteria must be fulfilled. One of these is that there must be a non-trivial second homotopy. Which i believe means that the resultant vacuum manifold must be non-trivial.
So for example if the vacuum manifold is a 2-sphere, the second homotopy classifies the ways you can map a 2-sphere onto this manifold. A 2-sphere cannot be deformed to a point, and thus we introduce winding numbers which can be associated with topological charge = magnetic monopole. [I may be wrong here]
So, in many papers discussing the electroweak monopole, the following statements appear in all of them:
'it was thought that the Weinberg-Salam model possesses no non-trivial second homotopy' (i.e. no monopoles exist)
'However, the Weinberg-Salam model with the hypercharge $U(1)$, could be viewed as a gauged $CP^1$ model in which the (normalized) Higgs doublet plays the role of the $CP^1$ field'
I confess that I am completely lost by this last statement. If anybody could shed any light as to what a gauged $CP^1$ field/$CP^1$ model is (or a good book that explains it) it would be great,