The Topological Invariants and Their Interrelations

Abstract

This chapter first reviews the cyclic cohomology for general C$^*$-algebras and its pairing with the K-theory, which produces numerical topological invariants. The discussion is then specialized to the algebras of physical observables. The strong and the weak topological invariants, for both bulk and boundary, are defined as pairings of specific cyclic cocycles with the elements of the K-groups encoding the topology of the solid state systems. The duality of the pairings with respect to the connecting maps is proved and the equality between the bulk and boundary invariants is established. Lastly, generalized Streda formulas are derived and used to determine the range of the topological invariants.