Classification of delicate and multigap topological bands protected by a combination of space inversion and additional discrete symmetries

Recently proposed delicate and multigap topological insulators (TIs) have considerably stronger restrictions on the band structure, rendering the conventional classification methods such as K-theory and topological quantum chemistry rather ineffective. However, systematic understanding of delicate and multigap TIs is of great importance in investigating exotic band properties such as large optical shift current, non-abelian braiding of band nodes, and multiply degenerate momenta. Although there is no known solution to the general classification of such unconventional topological bands, significant links to homotopy theory and classifying space theory has been observed. Building upon this, we employ the machinery of long exact sequence of homotopy groups and classify all delicate/multigap topological bands, in spatial dimensions zero to four, that are protected by the composite of space inversion $(\mathcal{I})$ and additional discrete symmetries from the Altland-Zirnbauer symmetry classes \red{$(\mathcal{T,P,C})$}. These composite symmetries \red{$(\mathfrak{T}\equiv \mathcal{TI},\, \mathfrak{P}\equiv \mathcal{PI}, \mathcal{C\equiv TP})$} are local in momentum space, enabling one to employ the machinery of homotopy theory to tackle the classification problem. As an application, we classify the band nodes whose braiding rule is determined by certain homotopy groups, and \red{show} that precisely two symmetry classes feature non-abelian braiding effect.