  We suggest that exotic sphere fibrations can be mapped to band topologies in
condensed matter systems. These fibrations can correspond to geometric phases
of two double bands or state vector bases with second Chern numbers m+n and -n
respectively. They can be related to topological insulators, magneto-electric
effects, and photonic crystals with special edge states. We also consider
time-reversal symmetry breaking perturbations of topological insulator, and
heterostructures of topological insulators with normal insulators and with
superconductors. We consider periodic TI/NI/TI/NI' heterostuctures, and
periodic TI/SC/TI/SC' heterostuctures. They also give rise to models of Weyl
semimetals which have thermal and electrical transports.
