A homotopical approach to symmetries of aperiodic patterns

Abstract

The translational structure of an aperiodically ordered pattern can be studied via the topology of an associated space, called the translational hull. Much can be said about these spaces ; for example there are now several techniques for computing their cohomology. However, information on rotational symmetry is lost by passing to the translational hull. To study this rotational structure topologically, we consider instead a more complicated space, the rotational hull of the pattern, denoted ${\Omega_{r}}$. This talk shall provide an introduction to the study of these spaces and discuss recent joint work with John Hunton which provides tools for studying the topologies of rotational hulls, for example in calculating their cohomology. In this work a space ${\Omega_{G}}$ was introduced, initially as an intermediate for studying ${\Omega_{r}}$. We show that it is perhaps more natural from a homotopical viewpoint by showing that for periodic patterns the fundemental group of ${\Omega_{G}}$ recovers the classical space group of Euclidean symmetries of the pattern. We thus introduce a new invariant, the pro- or space-fundamental group of ${\Omega_{G}}$, which extends the classical space group to aperiodic patterns. For certain cut and project patterns an alternative definition for the aperiodic space group was given by crystallographers. We compare the two notions by showing that our topological space group naturally projects to the aperiodic space group. The map is always non-injective and so our invariant appears to contain further information.