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

Martin Bendersky, Donald M. Davis, and Mark Mahowald

mbenders@shiva.hunter.cuny.edu
dmd1@lehigh.edu
mark@math.northwestern.edu

Abstract
In 1981, Davis, Gitler, and Mahowald determined the geometric dimension
of stable vector bundles of order 2^e over RP^{2n} if e > 74 and n is
sufficiently large.  In this paper, we use the Bendersky-Davis computation
of v1-periodic homotopy groups of 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>74 works for e>5, but for e \le 5 the
geometric dimension is often different due to anomalies in the v1-periodic
homotopy groups of SO(m) when m<11.