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Regularity of line defects in 3-dimensional media. (English) Zbl 0588.55012

A boundary value problem in the context of this work essentially is an extension problem of the following type: Given a 3-manifold M, a (defect) map into some parameter space V, defined on ”most” of the boundary \(\partial M\), is to be extended as a (defect) map on M. A ”regularly defect second homotopy group” serves as the coefficient group of a homology obstruction for boundary value problems enjoying certain regularity properties. The parameter space V is defined to have the regular defects property, if all such boundary value problems involving V have a regularly defect solution.
Theorem: If \(\Gamma\) \(\subset SO(3)\) is finite then \(V=S0(3)/\Gamma\) has the regular defects property. Investigating the regularly defect second homotopy group shows that the regular defects property depends only on the fundamental group of the parameter space; among other results, a necessary and sufficient condition for this property is found. A final paragraph discusses implications of this work for defects in ordered media in physics.
Reviewer: D.Erle

MSC:

55S36 Extension and compression of mappings in algebraic topology
55S40 Sectioning fiber spaces and bundles in algebraic topology
57M12 Low-dimensional topology of special (e.g., branched) coverings
55S35 Obstruction theory in algebraic topology
55Q70 Homotopy groups of special types
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
57R22 Topology of vector bundles and fiber bundles
57R20 Characteristic classes and numbers in differential topology
57M05 Fundamental group, presentations, free differential calculus
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