Symmetry defects and broken symmetry. Configurations Hidden Symmetry

This paper is an introduction to the study of spontaneous symmetry breaking and topological classification of defects. The latter topic has aits foundation in the former one; both subjects requires some mathematics not familiar to many physicists: group action and homotopy theory. These mathematics are introduced from examples and their main results, to be used for physics, are explained. It is hoped that this paper will enable the non specialist to read the physics literature which has very recently appeared on topological classification of symmetry defects. Some appendices (A,C,D,F,G) gives more mathematical details; the other appendices (B,E,H) explain some applications to different fields of physics, while mesomorphic phases are treated in the text (${\mathrm{He}}^{3}$ superfluid phases have been treated by Mermin, 1979).