The topology of non-uniform media in condensed matter physics

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Abstract
Methods of algebraic topology have been employed recently to classify defects and non-singular textures of condensed matter systems, and to describe two-defect processes. In this article a systematic review is presented of these methods and their applications. The non-uniform media are characterized by fields valued in a space of degeneracy, whose topological properties are investigated. Ways to represent such a space are reported. An introduction is given to the necessary mathematical tools, viz. the homotopy groups and exact sequences thereof. Combinations, entanglements and transformations of singularities are discussed. The connection between the homotopic defect classification and the description of many-defect systems is elaborated. The limitations and necessary modifications of the method for systems of broken translational symmetry are examined.

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