The stable geometric dimension of vector bundles over real projective spaces

An elementary argument shows that the geometric dimension of any vector bundle of order 2e over RP' depends only on e and the residue of n mod 8 for n sufficiently large. In this paper we calculate this geometric dimension, which is approximately 2e. The nonlifting results are easily obtained using the spectrum bJ. The lifting results require bo-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage bo-resolutions act almost like K(Z2)-resolutions.