BB-correspondence in solid state theory

A review of topological methods applied in solid-state theory is given. First, we recall the basic provisions of Bloch theory describing the properties of solids with a crystal lattice. Then we construct an algebra of observables of a topological dielectric and the resulting classes of symmetries and pseudosymmetries. Next, a description of the algebras of observables is given in terms of the $K$-theory of graded $C^*$-algebras and the accompanying topological invariants of a solid. The algebra of boundary observables is defined in terms of the $K$-theory proposed by Kasparov.
In conclusion, we describe the correspondence between the topological invariants of the body and its boundary (BB-correspondence). In the particular case of a periodic unitary model, this correspondence can be described explicitly.