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.
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 |