Quantum Principal Fiber Bundles: Topological Aspects

We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum bundle admits sections if and only if it is trivial. Using a quantum version of \v{C}ech cocycles, we obtain a reconstruction theorem for quantum principal bundles. The classification of bundles over a given quantum space as a base space is reduced to the corresponding problem, but with an ordinary classical group playing the role of structure group. Some explicit examples are considered.