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For a compact Hausdorff space X, the topological K-theory K(X) is defined to be the Grothendieck group of stable isomorphism classes of finite dimensional vector bundles over X. The complex or real K-group can be equivalently defined by homotopy classes of maps from X to the classifying spaces, K(X)=KC(X)≃[X,BU×Z], KO(X)=KR(X)≃[X,BO×Z]. In stable homotopy theory, we say the spectrum of the complex K-theory is BU ×Z, which has a period 2, i.e. Ω2BU ≃ BU × Z, where Ω is the loop space operation. Similarly, the spectrum of the real K-theory is BO × Z, which has a period 8, i.e. Ω8BO ≃ Z × BO. Such periodic phenomena, are known as the Bott periodicity. The first case was discovered by Bott in the study of homotopy groups of spheres using Morse theory. For unitary or orthogonal groups, Bott periodicity is spelled out as πk+1(BU) = πk(U) ≃ πk+2(U), πk+1(BO) = πk(O) ≃ πk+8(O). Besides in topological K-theory and homotopy groups of classical groups, Bott periodicity is also present in Clifford algebras and their representations. Of course, Clifford modules and topological K-theory are intimately related in index theory. Going back to band theory, in the language of K-theory via stabilizing (embed- ding into the direct limit) the nonlinear sigma model induces an element in the complex or real K-group, the homotopy class of the map Td →F Gr(N ) → BU (N ) → BU. [F] ∈ K(Td) for type A (5.6) KO(Td) for type AII. Furthermore, if the map from the Brillouin torus can transformed into a map Fˆ from the sphere Sn either by a quotient map or by extending and quotienting, as described in the previous section, then πn(BU) = πn−1(U) for type A (5.7) πn(BO) = πn−1(O) for type AII. [Fˆ] ∈ For instance, consider the edge Hamiltonian in the quantum spin Hall effect, i.e. n = 1, then the 2d Z2 invariant corresponds to π1(BO) = π0(O) ≃ Z2. 5.3. Clifford modules Following [51], we briefly go over the classification approach via the Clifford exten- sion problem in this subsection. For this one regards free fermions in the basis of operators cˆn which satisfy the Clifford relations cˆlcˆm + cˆmcˆl = 2δl,m. 1630003-47 Notes on topological insulators Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
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