Topology of Misorientation Spaces

Let $G_1$ and $G_2$ be finite subgroups of $\mathrm {SO}(3)$. The double quotients of the form $X(G_1,G_2)=G_1\backslash\mathrm{SO}(3)/G_2$ 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 $3$-manifolds with finite fundamental groups. 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 appealing to Perelman's theorem. The classification of misorientation spaces yields new $n$-valued group structures on the manifolds $S^3$ and $\mathbb R\mathrm P^3$. Finally, we outline the connection of the particular misorientation space $X(D_2,D_2)$ with integrable dynamical systems and toric topology.