Ammann tilings in symplectic geometry (Q352482)

The simple or primitive icosahedral quasilattice \(P\) is generated by vectors pointing to the twelve vertices of an icosahedron. The primitive icosahedral tiling \(T\), whose vertices lie in \(P\), was introduced independently by R. Ammann [\textit{A. L. Mackay} ``De nive quinquangula: on the pentagonal snowflake'', Kristallografiya, 26, 909--918 (1981); Soviet Physics Crystallography, 26, No. 5, 910--919 (1981)] and by \textit{P. Kramer} and \textit{R. Neri} [Acta Cryst. A 40, No. 5, 580--587 (1984; Zbl 1176.52010)]. The two building blocks of \(T\) are the oblate and prolate Ammann rhombohedra.NEWLINENEWLINEA generalization of a construction by \textit{T. Delzant} [Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)], which associates a symplectic toric \(2n\)-manifold to each simple convex polytope in \((\mathbb R^n)^{\ast}\), is used to construct a symplectic space for each tile. The resulting space is a \(2n\)-dimensional quasifold, a generalization of manifolds and orbifolds [\textit{E. Prato}, Topology 40, No. 5, 961--975 (2001; Zbl 1013.53054); the authors, Commun. Math. Phys. 299, No. 3, 577--601 (2010; Zbl 1205.53087)].NEWLINENEWLINEA symplectic quasifold \(M_{b}\) is associated to each of the oblate rhombohedra of the tiling with fixed edge length, and another symplectic quasifold \(M_{r}\) is associated to each of the prolate rhombohedra. \(M_{b}\) and \(M_{r}\) are globally the quotient of the product of three 2-spheres modulo the action of a discrete group. It is proved that \(M_{b}\) and \(M_{r}\) are diffeomorphic but not symplectomorphic, a property shared by the quasifolds appearing in the study of the Penrose rhombus tiling from the point of view of symplectic geometry.