  We study the application of Kasparov theory to topological insulator systems
and the bulk-edge correspondence. We consider observable algebras as modelled
by crossed products, where bulk and edge systems may be linked by a short exact
sequence. We construct unbounded Kasparov modules encoding the dynamics of the
crossed product. We then link bulk and edge Kasparov modules using the Kasparov
product. Because of the anti-linear symmetries that occur in topological
insulator models, real $C^*$-algebras and $KKO$-theory must be used.
