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The topological classification of gapped band structures depends on theparticular definition of topological equivalence. For translation-invariantsystems, stable equivalence is defined by a lack of restrictions on the numbersof occupied and unoccupied bands, while imposing restrictions on one or bothleads to ``fragile'' and ``delicate'' topology, respectively. In this article,we describe a homotopic classification of band structures -- which captures thetopology beyond the stable equivalence -- in the presence of additional latticesymmetries. As examples, we present complete homotopic classifications forspinless band structures with twofold rotation, fourfold rotation and fourfolddihedral symmetries, both in presence and absence of time-reversal symmetry.Whereas the rules of delicate and fragile topology do not admit a bulk-boundarycorrespondence, we identify a version of stable topology, which restricts therepresentations of bands, but not their numbers, which does allow for anomalousstates at symmetry-preserving boundaries, which are associated with nontrivialbulk topology.

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