The geometric dimension of some vector bundles over projective spaces

Abstract:

We prove that in many cases the geometric dimension of the $p$-fold Whitney sum $p{H_k}$ of the Hopf bundle ${H_k}$ over quaternionic projective space $Q{P^k}$ is the smallest $n$ such that for all $i \leq k$ the reduction of the $i$th symplectic Pontryagin class of $p{H_k}$ to coefficients ${\pi _{4i - 1}}(({\text {R}}{P^\infty }/{\text {R}}{P^{n - 1}})\Lambda bo)$ is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space ${\text {R}}{P^{4k + 3}}$ in Euclidean space if the number of 1’s in the binary expansion of $k$ is between 5 and 8.