Elementary Classification of Topological Insulators in Low Dimensions

Topological insulators can be classified depending on the symmetries and dimension of the physical system. The classification often involves advanced mathematical tools. The goal of this thesis is to understand the classification in low dimensions using elementary tools from topology and differential geometry. Our direct approach without relying on the bulk-boundary correspondence makes the classification more accessible to students who are new to the subject. We classify the topological insulators via homotopy theory. For each symmetry class in dimension 0, 1 and 2, we either define an index in terms of equivariant vector bundles to distinguish between different homotopy classes or we show that there is only one homotopy class. For the Z 2 -indices, we discuss how they are related to the higher dimensional Z or Z 2 -index. Moreover, we provide examples to show that the indices defined are indeed non-trivial.