  We study topological states of matter in quasicrystals, which do not rely on
crystalline orders. In the absence of a bandstructure description and
spin-orbit coupling, we show that a three-dimensional quasicrystal can
nevertheless form a topological insulator. It relies on a combination of
noncrystallographic rotational symmetry of quasicrystals and electronic orbital
space symmetry, which is the quasicrystalline counterpart of topological
crystalline insulator. The resulting topological state obeys a non-trivial
twisted bulk-boundary correspondence and lacks a good metallic surface. The
topological surface states, localized on the top and bottom planes respecting
the quasicrystalline symmetry, exhibit a new kind of multifractality with
probability density concentrates mostly on high symmetry patches. They form a
near-degenerate manifold of 'immobile' states whose number scales
proportionally with the macroscopic sample size. This can open the door to a
novel platform for topological surface physics distinct from the crystalline
counterpart.
