  We describe explicit generators for the "real" K-theory of "real" spheres in
van Daele's picture. Pulling these generators back along suitable maps from
tori to spheres produces a family of Hamiltonians used in the physics
literature on topological insulators. We compute their K-theory classes
geometrically, based on wrong-way functoriality of K-theory and the geometric
version of bivariant K-theory, which we extend to the "real" case.
