  We show that, for each symmetry class based on the tenfold way
classification, the effective Dirac operator obtained by integrating out the
additional bulk direction takes a value in the corresponding classifying space,
from which we obtain the flat band Hamiltonian. We then obtain the overlap
Dirac operator for each symmetry class and establish the Ginsparg--Wilson
relation associated with $\mathcal{C}$ and $\mathcal{T}$ symmetries, and also
the mod-two index theorem.

High Energy Physics - Theory;Mesoscale and Nanoscale Physics;High Energy Physics - Lattice;Differential Geometry