ON EXOTIC SPHERE FIBRATIONS, TOPOLOGICAL PHASES, AND EDGE STATES IN PHYSICAL SYSTEMS

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, magnetoelectric 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′ heterostructures, and periodic TI/SC/TI/SC′ heterostructures. They also give rise to models of Weyl semimetals which have thermal and electrical transports.

On a distance function between Differentiable structures

In this note, we investigate a relation between the connected sum of manifolds and the distance of manifolds ([2]). Since the smoothing of a piecewise linear equivalence is given by connected sum of exotic spheres ([1]), we have a certain estimate of the smoothing obstruction using the distance of manifolds (Proposition 3). In § 3, an application is given to show the impossibility of the 0.64-pinching of an exotic sphere.