Topology of misorientation spaces

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 $\pi_1(X(G_1,G_2))$, and apply Thurston's elliptization conjecture to describe these spaces. Many misorientation spaces are homeomorphic to $S^3$ by Poincar\'{e} 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 Poincar\'{e} 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.