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Topology of misorientation spaces

Anton Ayzenberg, Dmitry Gugnin

Abstract

Let $G_1$ and $G_2$ be discrete subgroups of $SO(3)$. The double quotients of the form $X(G_1,G_2)=G_1\backslash SO(3)/G_2$ 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 $G_1,G_2$ are crystallography groups, we compute the fundamental groups $π_1(X(G_1,G_2))$, and apply Thurston's elliptization conjecture to describe these spaces. Many misorientation spaces are homeomorphic to $S^3$ 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 $S^3$ and $\mathbb{R}P^3$. Finally, we outline the connection of the particular misorientation space $X(D_2,D_2)$ to integrable dynamical systems and toric topology.

Topology of misorientation spaces

Abstract

Let and be discrete subgroups of . The double quotients of the form 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 are crystallography groups, we compute the fundamental groups , and apply Thurston's elliptization conjecture to describe these spaces. Many misorientation spaces are homeomorphic to 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 -valued group structures on and . Finally, we outline the connection of the particular misorientation space to integrable dynamical systems and toric topology.

Paper Structure

This paper contains 13 sections, 20 theorems, 36 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Misorientation spaces
  3. Finite group actions on 3-manifolds
  4. Misorientation spaces and double covers
  5. Rotations in $\mathop{\mathrm{SO}}\nolimits(4)$
  6. Coordinates on several misorientation spaces
  7. $n$-valued topological groups
  8. Coordinates given by consecutive folding
  9. Algebraical coordinates given by invariants
  10. Several instances of $X(D_2,D_2)$
  11. Real moment map
  12. Periodic Toda lattice
  13. The argument with the ring of invariants

Key Result

Proposition 1.2

$\widetilde{X}(G)$ is homeomorphic to the 3-dimensional disc $D^3$ if $G$ belongs to the list: $\{D_2,D_4,D_6,T,O\}$. $\widetilde{X}(G)$ is homeomorphic to the cone over real projective plane $\mathop{\mathrm{Cone}}\nolimits \mathbb{R}P^2$ if $G$ belongs to the list $\{C_2,C_3,C_4,C_6,D_3\}$.

Figures (1)

  • Figure 1: The subdivision of $\widetilde{p}^{-1}(0)/\mathbb{Z}_2^3\cong T^2$ into strata $\{b_1=0\}$, $\{b_2=0\}$, and $\{b_3=0\}$

Theorems & Definitions (46)

  • Definition 1.1
  • Proposition 1.2: Patala and Schuh PatSch
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5: Lange--Mikhailova
  • Proposition 2.1
  • Proposition 2.2: Mikhailova, Mikh
  • Proposition 2.3: Armstrong, Arm
  • Corollary 2.4
  • Remark 2.5
  • ...and 36 more