The crossing of defects in ordered media and the topology of 3-manifolds

2014 This investigation is a first step in the study of the topological obstructions involved in deforming the defects of ordered media; it belongs to the framework of the recent physical theories which classify stable defects in terms of the homotopy groups of a certain manifold V, characteristic for the given type of order. In particular, it is shown here that the only obstruction for having two defect lines in a 3-dimensional sample cross through each other, without getting entangled, is a certain commutator in the fundamental group of V. This represents a qualitatively new phenomenon as far as the behaviour of certain materials (with non-commutative 03C01 V), some still to be synthesized, is concerned ; it also asks for a revision of certain traditional concepts in the physical theory of condensed matter. The present paper contains a rigorous mathematical framework for the description of non-commutative defects, a discussion of some physical applications and some open problems. Tome 38 N° 8 AOUT 1977 LE JOURNAL DE PHYSIQUE