Quantum principal fibre bundles: Topological aspects

We introduce the notion of locally trivial quantum principal bundles. The base space and the total space are compact quantum spaces (unital C∗-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 Čech cocycles, we obtain a reconstruction theorem for the quantum principal bundles. The classification of bundles over a given quantum space as the base space is reduced to the corresponding problem with an ordinary classical group playing the role of the structure group. Some explicit examples are considered.