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We suggest that exotic sphere fibrations can be mapped to band topologies incondensed matter systems. These fibrations can correspond to geometric phasesof two double bands or state vector bases with second Chern numbers m+n and -nrespectively. They can be related to topological insulators, magneto-electriceffects, and photonic crystals with special edge states. We also considertime-reversal symmetry breaking perturbations of topological insulator, andheterostructures of topological insulators with normal insulators and withsuperconductors. We consider periodic TI/NI/TI/NI' heterostuctures, andperiodic TI/SC/TI/SC' heterostuctures. They also give rise to models of Weylsemimetals which have thermal and electrical transports.

On exotic sphere fibrations, topological phases, and edge states in physical systems