## Stable geometric dimension of vector bundles over even-dimensional real projective spaces

Martin Bendersky, Donald M. Davis, and Mark Mahowald

This 23-page paper will appear in Transactions of the American Math Society in 2005.

Abstract
In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles or order 2^e over RP^{2n} if n is sufficiently large and e \ge 75. In this paper, we use the Bendersky-Davis computation of v1^{-1}pi_*(SO(m)) to determine this geometric dimension for all values of e (still provided that n is sufficiently large). The same formula that worked for e \ge 75 works for e \ge 6, but for e<6 the formula is different due to anomalies in the formula for v1^{-1}pi_*(SO(m)) when m \le 10.