We introduce the cluster exchange groupoids to show that any connected component (which are all isomorphic) of the moduli space of framed quadratic differentials on a decorated marked surface S_Delta is simply connected. We also improve the result of Bridgeland-Smith, that such a space can be identified with the principal component of the space of stability conditions on the associated Calabi-Yau-3 categories of S_Delta. (cf. Link)