Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an elementary band representation (atomic limit band) splits into a particular set of bands. For the first time, we obtain a complete classification of the fragile topological phases, which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure that far surpasses that of stable or strong topological states. We find and enumerate hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues, and we link the mathematical structure of these phases to that of affine monoids in mathematics. Furthermore, for the first time, we predict and calculate (hundreds of realistic) materials where fragile topological bands appear, and we showcase the very best ones.
