  In this paper, we propose a new type of bulk-boundary correspondence as a
generic approach to theoretically and experimentally detect fragile topological
states. When the fragile phase can be written as a difference of a trivial
atomic insulator and the so-called obstructed atomic insulator, the gap between
the fragile phase and other bands must close under a specific novel twist of
the boundary condition of the system. We explicitly work out all the twisted
boundary conditions (TBC) that can detect all the 2D fragile phases implied by
symmetry eigenvalues in all wallpaper groups. We develop the concept of real
space invariants - local good quantum numbers in real space - which fully
characterize the eigenvalue fragile phases. We show that the number of
unavoidable level crossings under the twisted boundary condition is completely
determined by the real space invariants. Possible realizations of the TBC of
the fragile band in metamaterial systems are discussed.
