Topology of Misorientation Spaces

Let and be finite subgroups of . The double quotients of the form were introduced in materials science under the name misorientation spaces. In this paper we review several known results that allow one to study the topology of misorientation spaces. Neglecting the orbifold structure, one can say tha...

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Bibliographic Details
Published in:Proceedings of the Steklov Institute of Mathematics 2024-06, Vol.325 (1), p.1-20
Main Authors: Ayzenberg, Anton A., Gugnin, Dmitry V.
Format: Article
Language:English
Subjects:
14/34
639/766/189
639/766/530
639/766/747
Crystallography
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Misalignment
Subgroups
Theorems
Topology
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Summary:Let and be finite subgroups of . The double quotients of the form were introduced in materials science under the name misorientation spaces. In this paper we review several known results that allow one to study the topology of misorientation spaces. Neglecting the orbifold structure, one can say that all misorientation spaces are closed orientable topological -manifolds with finite fundamental groups. In the case when and are crystallographic groups, we compute the fundamental groups and apply the elliptization theorem to describe these spaces. Many misorientation spaces are homeomorphic to by Perelman’s theorem. However, we explicitly describe the topological types of several misorientation spaces without appealing to Perelman’s theorem. The classification of misorientation spaces yields new -valued group structures on the manifolds and . Finally, we outline the connection of the particular misorientation space with integrable dynamical systems and toric topology.
ISSN:0081-5438
1531-8605
DOI:10.1134/S0081543824020019