Topology behind Topological Insulators

In this paper topological K-group calculations for fiber bundles with structure group SO(3) over tori are carried out to explain why topological insulators have special conducting points on their surface but are bulk insulators. It is shown that these special points are gapless and conducting for topological reasons and follow from the K-group calculations. The existence of gapless surface points is established with the help of an additional topological property of the K-groups which relates them to the index theorem of an operator. The index theorem relates zeros of operators to topology. For the topological insulator the relevant operator is a Dirac operator, that emerges in the problem because the system has strong spin-orbit interactions and time reversal invariance. Calculating K-groups over tori require some special topological tools that are not widely known. These are explained. We then show that the actual calculation of K-groups over tori becomes straightforward once a few topological results are in place. Since condensed matter systems with periodic lattices are always bundles over tori the procedures described is of general interest.