Introduction to topological insulators from the viewpoint of topology

This is an expository article about topological insulators from the viewpoint of topology: We introduce a mathematical model of topological insulators and explain their classification by topological K K -theory. In the presence of symmetries, the classification is generalized by using twisted equivariant K K -theory, and the periodic table of topological insulators and superconductors is derived. Some results about K K -theoretic classifications of topological crystalline insulators are also reviewed.