When studying topological defects, the parameter space (or vacuum manifold in QFT) is denoted as the coset space $G/H$, where $G$ corresponds to the symmetry group of the Lagrangian and $H$ the remaining symmetry after some spontaneous symmetry breaking. Now usually $G$ is not unique in the sense that it can be chosen arbitrarily large or small (e.g. $SO(3)$ or $O(3)$ are viable options for 3 dim. spin systems). The physically sensible argument is now to say that $G/H$ is unique, i.e. $G/H \cong G'/H' $ for any other choice of $G'$ (and its corresponding $H'$), because any differences "quotient out anyway". I have not yet found any rigorous mathematical footing of this statement and want to ask, if you know of any ?
(this property becomes particularly handy, if one wants to compute their homotopy groups and thus want to choose the universal covering group for the initial choice of $G$..., so maybe the question should be why $G/H \cong \widetilde{G}/\widetilde{H} $, where $\widetilde{G}$ is the universal cover of $G$)
