DEFINABLE CrFIBER BUNDLES AND DEFINABLE CrG VECTOR BUNDLES

Abstract

Let G and K be compact subgroups of orthogonal groups and <TEX>$0{\leq}r</TEX><TEX><</TEX><TEX>x</TEX><TEX><</TEX><TEX>{\infty}$</TEX>. We prove that every topological fiber bundle over a definable <TEX>$C^r$</TEX> manifold whose structure group is K admits a unique strongly definable <TEX>$C^r$</TEX> fiber bundle structure up to definable <TEX>$C^r$</TEX> fiber bundle isomorphism. We prove that every G vector bundle over an affine definable <TEX>$C^rG$</TEX> manifold admits a unique strongly definable <TEX>$C^rG$</TEX> vector bundle structure up to definable <TEX>$C^rG$</TEX> vector bundle isomorphism.