  The rapid development of topological photonics and acoustics calls for
accurate understanding of band topology in classical waves, which is not yet
achieved in many situations. Here, we present the Wilson-loop approach for
exact numerical calculation of the topological invariants for several
photonic/sonic crystals. We demonstrate that these topological photonic/sonic
crystals are topological crystalline insulators with fragile topology, a
feature which has been ignored in previous studies. We further discuss the
bulk-edge correspondence in these systems with emphasis on symmetry broken on
the edges.
