On Stable Vector Bundles over Real Projective Spaces

If X is a connected, finite CJF-complex, we can define iKO)~iX) to be [X, BO] (base-point preserving homotopy classes of maps). Recall [2] that if xEiKO)~iX), the geometrical dimension of x (abbreviated g.dim x) can be defined to be the smallest nonnegative integer k such that a representative of x factors through BO(k). If $ is a vector bundle over X, the class in (PO)~(X) of a classifying map for £ is called the stable class of £. Let P" denote real projective ra-space, and let x„G(PC)~(Pn) denote the stable class of the canonical line bundle yn over P". If ?ra is a positive integer, let im) denote the number of integers k such that 0<k^m and k = 0, 1, 2, 4 (mod 8). The purpose of this note is to prove the following theorem: