Topology of misorientation spaces (Q6633145)

``Let \(G_1\) and \(G_2\) be finite 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 materials science under the name misorientation spaces. In this article, we review several known results that allow one to study the topology of misorientation spaces.''\N\N``Definition 1.1: The topological orbit space \(X(G_1,G_2) = SO(3)/(G_1 \times G_2) = G_1 \backslash SO(3)/ G_2\) is called the \textit{misorientation space} of the pair of groups \(G_1\), \(G_2\)'', where the two-sided actions on \(SO(3)\) are defined by \((g_1, g_2)A = g_1 \cdot A \cdot g_2\) for \((g_1, g_2) \in G_1 \times G_2\) and \(A \in SO(3)\).\N\N``In the case when \(G_1\) and \(G_2\) are crystallographic groups, we compute the fundamental groups \(\pi_1(X(G_1,G_2))\) and apply the elliptization theorem to describe these spaces. Many misorientation spaces are homeomorphic to \(S^3\) by Perelman's theorem. However, we explicitly describe the topological types of several misorientation spaces without relying on Perelman's theorem.''\N\NThe full classification of finite subgroups of \(SO(3)\) is known. These are cyclic, dihedral, tetrahedral, octahedral and icosahedral groups which are respectively denoted by \( \mathbb Z_m\), \(\hbox{D}_m\), \(T\), \(O\) and \(I\). The misorientation spaces \(X(G_1,G_2)\) have been studied well if one of the acting groups is trivial. In this case, the remaining action, say \(G_1\), by the left multiplication on \(SO(3)\) is free, thus the orbit space is an elliptic manifold whose universal covering space is \(S^3 \cong Sp(1)\), which is the double cover of \(SO(3)\).\N\NPossible manifolds for \(X(G_1, 1)\) are:\N\begin{itemize}\N\item If \(G_1\) is also trivial, then \(X(1, 1)\) is \(SO(3) \cong \mathbb R\hbox{P}^3\).\N\item If \(G_1 = \mathbb Z_m\), then \(X(\mathbb Z_m, 1)\) is the lens space \(L(2m, 1) \cong S^3/\mathbb Z_{2m}\).\N\item If \(G_1 = \hbox{D}_2 = Z_2 \times \mathbb Z_2\), then \(X(\hbox{D}_2, 1)\) is the quaternionic manifold \(S^3/Q_8\), where \(Q_8 = \{\pm 1, \pm i, \pm j, \pm j\}\).\N\end{itemize}\NIn this paper, the authors thus rather consider the case where both groups \(G_1\) and \(G_2\) are non-trivial. If the two-sided action \(G_1 \times G_2\) on \(SO(3)\) is not free, meaning there is a fixed point under the action, then \(X(G_1,G_2)\) becomes an elliptic orbifold (see [\textit{M. Boileau} et al., Ann. Math. (2) 162, No. 1, 195--290 (2005; Zbl 1087.57009)]). According the authors, there seems to no precise description of the underlying topology of these orbifolds in the literature. In this sense, this article fills in this gap, and some of the main results are as follows.\N\NTheorem.\NThe misorientation space \(X(G, G)\) is homeomorphic to\N\begin{itemize}\N\item[(1)] the three-sphere \(S^3\) if \(G \in \{ T, O, I \}\).\N\item[(2)] \(\mathbb R \hbox{P}^3\) if \(G = \mathbb Z_n\) for all \(n \geq 2\).\N\item[(3)] \(\mathbb R \hbox{P}^3\) if \(G = \hbox{D}_n\) for all odd number \(n\).\N\item[(4)] the three-sphere \(S^3\) if \(G = \hbox{D}_n\) for all even number \(n\).\N\end{itemize}\N\NMoreover, some of the remaining cases are: \(X(\mathbb Z_2, \mathbb Z_3) \cong L(12, 5)\); \(X(\mathbb Z_2, \mathbb Z_4) \cong L(4, 1)\); \(X(\mathbb Z_2, \mathbb Z_6) \cong L(6, 1)\); \(X(\mathbb Z_2, T) \cong L(3, 1)\); \(X(\mathbb Z_2, \hbox{D}_3) \cong \mathbb R \hbox{P}^3\); \(X(\mathbb Z_2, G_2) \cong S^3\) for \(G \in \{\hbox{D}_2, \hbox{D}_4, \hbox{D}_6, O, I\}\). If \(X(\mathbb Z_3, \hbox{D}_n)\) for \(n = 2\), \(4\), then it is ``free'' meaning a manifold obtained by taking the quotient by an acting group \(\mathbb Z_3\) on the quaternionic manifold. If \(G_1 \in \{\hbox{D}_3, \hbox{D}_4, \hbox{D}_6, T, O, I \}\) and \(G_2 \in \{\hbox{D}_2, \hbox{D}_4, \hbox{D}_6, T, O, I \}\) for \(G_1 \neq G_2\), then \(X(G_1,G_2) \cong S^3\) except \(X(\hbox{D}_4, T) \cong L(3, 1)\).\N\NIt should be remarked that elliptic orbifolds whose underlying spaces homeomorphic to \(S^3\) are extensively studied by \textit{W. D. Dunbar} in [Rev. Mat. Univ. Complutense Madr. 1, No. 1--3, 67--99 (1988; Zbl 0655.57008)], thus readers are encouraged to read this article as well.