Comparison of stratified-algebraic and topological K-theory

Stratied-algebra ic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more exible. We give a characterization of the compact real algebraic varieties X having the following property: There exists a positive integer r such that for any topological vector bundle on X, the direct sum of r copies of is isomorphic to a stratied- algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.