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We propose a real-space formalism of the topological Euler class, whichcharacterizes the fragile topology of two-dimensional systems with real wavefunctions. This real-space description is characterized by local Euler markerswhose macroscopic average coincides with the Euler number, and it appliesequally well to periodic and open boundary conditions for both crystals andnoncrystalline systems. We validate this by diagnosing topological phasetransitions in clean and disordered crystalline systems with the realityendowed by the space-time inversion symmetry $\mathcal{I}_{ST}$. Furthermore,we demonstrated the topological Euler phases in quasicrystals and even inamorphous lattices lacking any spatial symmetries. Our work not only provides alocal characterization of the fragile topology but also significantly extendsits territory beyond $\mathcal{I}_{ST}$-symmetric crystalline materials.

Real-space Formalism for the Euler Class and Fragile Topology in Quasicrystals and Amorphous Lattices