Topology, manifolds, and homotopy: Basic concepts and applications to $n$-field models

The basic concepts of topology, the theory of manifolds, and the theory of homotopy, which are employed in the study of extended objects both in field theory and in the physics of the condensed state, are presented. The physical systems (magnetic materials and nematics) are described first. Their topological properties illustrate the general topological concepts which are introduced later. Thus two goals are pursued. On the one hand, a detailed analysis of examples completes the brief exposition of the general mathematical questions, while on the other the physical systems automatically become objects of topological study. In particular, the topological defects in the field of the order parameter of ordered systems are classified with the help of homotopic groups.