Finite volume calculation of K-theory invariants

Odd index pairings of K1-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finitedimensional matrix, more precisely the finite volume restriction of what we call the spectral localizer. In presence of real symmetries, secondary Z2-invariants can be obtained as the sign of the Pfaffian of the spectral localizer. These results reconcile two complementary approaches to invariants of topological insulators.