efficient way to study these properties. Each group π$_{n}$ ( M ) is composed of all classes of hypersurfaces of dimension n that can shrink to a point while staying inside M [458]. If any hypersurface can shrink to a point, the homotopy group contains only one element and is said to be trivial. In particular, π$_{0}$ ( M ) is trivial if and only if M is connected, π$_{1}$ ( M ) is trivial if and only if M is simply connected. One can easily visualize that if M is not connected (for example during the breaking of a discrete group Z$_{n}$ → I ), uncorrelated regions of the universe will fall in different vacua and will necessarily be separated by domain walls [449, 450]. With the same reasoning, if the universe undergo a phase transition satisfying π$_{1}$ [ G/H ] = I , cosmic strings , that is line-like defects will necessarily form, with a density given by the correlation length or the mass scale of the Higgs field responsible for the symmetry breaking. It is for example the case for U (1) → I or all symmetries of the type G → H × Z$_{n}$ . In general, the formation of topological defects of space-time dimension d is governed by the non-triviality of the homotopy group: One can show that any symmetry breaking of the form G → H × U (1) gives rise to the formation of monopole (point-like defects). This is the origin of the well-known monopole problem, since the Standard Model group contains a U (1) factor. This formation of unwanted defects was one of the original motivation to introduce a phase of inflation [3]. Note that the above topological conditions of formation of defects only govern the for- mation of topologically stable defects. It was however found that defects solutions can form even when the topology is trivial [451, 453]. The most well-known example are the electro- weak strings, formed during the eletroweak symmetry breaking which are perturbatively stable for a range of parameters which are not realized in nature, and belong to the broader class of embedded defects. These defects are a priori unstable though mechanisms (such as plasma effects) have been found to stabilize them. They are of interest for inflation model builders since this mechanism can allow lift the constraints from the formation of cosmic strings (see Sec. IV F on D -term inflation) $^{π$_{3}$}$− d = I . (156) 72