Applications of homotopy theory to topologicalsolitons

Abstract

This thesis is an introduction to topological the topological theory defects and solitons, aimed tobring the mathematical and physical perspectives closer together.The ordered media in which defects arise are introduced, and their configurations are describedusing homotopy theory. This description is reformulated in terms of symmetry groups, isotropygroups and homogeneous spaces, of which the homotopy groups are calculated using the longexact sequence of homotopy groups for fibrations. Finally, an overview of the applications of thismethod to a variety of systems is given, and the secondary phenomena arising due to the presenceof defects or solitons are described, concluding with the complete classification of defects inSO(3)symmetry broken media in 3 dimensions.