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Topology of misorientation spacesстатья
- Авторы: Ayzenberg Anton, Gugnin Dmitry
- Сборник: ArXiv
- Серия: Series arxive "math"
- Том: 1912
- Год издания: 2019
- Издательство: Cornell University Press
- Местоположение издательства: United States
- Первая страница: 1
- Последняя страница: 24
- DOI: 10.48550/arXiv.1912.11324
- Аннотация: Let G1 and G2 be discrete subgroups of SO(3). The double quotients of the form X(G1,G2)=G1∖SO(3)/G2 were introduced in material science under the name misorientation spaces. In this paper we review several known results that allow to study topology of misorientation spaces. Neglecting the orbifold structure, all misorientation spaces are closed orientable topological 3-manifolds with finite fundamental groups. In case when G1,G2 are crystallography groups, we compute the fundamental groups π1(X(G1,G2)), and apply Thurston's elliptization conjecture to describe these spaces. Many misorientation spaces are homeomorphic to S3 by Poincaré conjecture. The sphericity in these examples is related to the theorem of Mikhailova--Lange, which constitutes a certain real analogue of Chevalley--Shephard--Todd theorem. We explicitly describe topological types of several misorientation spaces avoiding the reference to Poincaré conjecture. Classification of misorientation spaces allows to introduce new n-valued group structures on S3 and RP3. Finally, we outline the connection of the particular misorientation space X(D2,D2) to integrable dynamical systems and toric topology.
- Добавил в систему: Гугнин Дмитрий Владимирович