Equivariant semialgebraic vector bundles

Abstract Let G be a compact semialgebraic group. We prove that any semialgebraic G -vector bundle over a semialgebraic G -set has a semialgebraic classifying G -map. Moreover we prove that the set of semialgebraic G -isomorphism classes of semialgebraic G -vector bundles over a semialgebraic G -set corresponds bijectively to the set of topological G -isomorphism classes of topological G -vector bundles over it. As an application of them, we prove that for any two semialgebraic G -maps f , h between semialgebraic G -sets M and N , if they are G -homotopic and ξ is a semialgebraic G -vector bundle over N , then f ∗ (ξ) and h ∗ (ξ) are semialgebraically G -isomorphic.