Stable Geometric Dimension of Vector Bundles over Odd-dimensional Real Projective Spaces

In [6], the geometric dimension of all stable vector bundles over real projective space P was determined if n is even and sufficiently large with respect to the order 2 of the bundle in K̃O(P). Here we perform a similar determination when n is odd and e > 6. The work is more delicate since P does not admit a v1-map when n is odd. There are a few extreme cases which we are unable to settle precisely. 1. Statement of results The geometric dimension gd(θ) of a stable vector bundle θ over a space X is the smallest integer m such that θ is stably equivalent to an m-plane bundle. Equivalently, gd(θ) is the smallest m such that the classifying map X θ −→ BO factors through BO(m). The group K̃O(P ) of equivalence classes of stable vector bundles over real projective space is a finite cyclic 2-group generated by the Hopf line bundle ξn. In [6], it was shown that, for sufficiently large even n, the geometric dimension of a stable vector bundle over P n depends only on its order in K̃O(P ) and the mod 8 value of n. For bundles of order 2, this value, called sgd(n, e) or sgd(n, e), where n is the mod 8 residue of n, was completely determined; its approximate value is 2e. A key role in this analysis was played by KO-equivalences P n+8 k+8 → P n k , defined if n is even, k is odd, and n + 8 < 2k − 1. Such maps do not exist when n is odd, and so the methods and results are somewhat more complicated. The term “stable” geometric dimension (sgd) refers to the fact that the geometric dimension achieves a stable value as n gets large within its congruence class. Date: June 14, 2005. 1991 Mathematics Subject Classification. 55S40,55R50,55T15.