Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology

Fragile and delicate topological phases of noninteracting fermions can be trivialized under addition of trivial bands, in contrast to strong topological phases. Here, the authors describe a classification procedure based on homotopy theory for topological insulators with a fixed number of bands. They thereby classify band structures symmetric under discrete rotations and time-reversal at the highest possible level of granularity and identify instances of ``representation-protected stable topology'', which is robust to addition of trivial bands with a fixed orbital type.