The local-triviality dimension of actions of compact quantum groups

We define the local-triviality dimension for actions of compact quantum groups on unital C*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when this dimension is finite. For commutative C*-algebras, this notion recovers the standard definition of local triviality of compact principal bundles. We prove that actions with finite local-triviality dimension are automatically free. Then we apply this new notion to prove the noncommutative Borsuk-Ulam-type conjecture under the assumption that a compact quantum group admits a non-trivial classical subgroup whose induced action has finite local-triviality dimension. This is a noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial principal bundles.