DEFINABLE G CW COMPLEX STRUCTURES OF DEFINABLE G SETS AND THEIR APPLICATIONS

Let G be a compact definable group. We prove that every pair of a definable G set and its closed definable G subset admits simultaneously definable G CW complex structures. As its applications, we prove that a canonical map from the set of definable G homotopy classes of definable G maps between definable G sets to that of G homotopy classes of continuous G maps between them is bijective. Moreover we prove that if G is a finite group, then the set of G vector bundle isomorphism classes of G vector bundles over a definable G set corresponds bijectively to that of definable G vector bundle isomorphism classes of definable G vector bundles.