Index Theorems for Solid State Systems
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The values of the parings between cyclic cohomology and K-theory have already been determined in Sect. 5.7 and only the strong topological invariants take integer values in general. In this chapter index theorems are proved for these strong invariants which allow to extend this integrality to the regime with mobility gap. For this purpose it is first shown how cyclic cocycles obtained from Fredholm modules pair integrally with K-theory. When combined with a local index formula, this allows to prove the integrality of the strong topological invariants and to establish their stability under MBGH. Furthermore, the delocalized nature of surface states is proved for non-trivial topological insulators.