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Abstract

These notes present an introduction to Noncommutative Topology by the example of compact quantum surfaces. First it is shown that the Toeplitz algebra may be viewed as the C$^*$-algebra of continuous functions on the quantum disk. The Toeplitz algebra gives rise to a C$^*$-algebra extension of the C$^*$-algebra of continuous functions on the boundary circle by the closed ideal of compact operators on a separable Hilbert space. The quantum surfaces are then defined by prescribing boundary conditions on the circle that correspond in the classical case to gluing pairs of arcs. The classification of isomorphism classes will be achieved by using Brown–Douglas–Fillmore theory on the classification of essentially normal operators. In contrast to the classical case, the isomorphism classes of non-orientable closed quantum surfaces depend on a parameter that counts the number of arcs that are identified along the same orientation. However, the topological invariants in the form of K-groups don’t change, that is, the K-groups of the closed quantum surfaces are isomorphic to their classical counterparts. Finally, explicit generators of these K-groups are given. For the convenience of the reader, these notes contain a brief introduction to the K-theory of C$^*$-algebras.