A stability theorem for the index of sphere bundles

Abstract

We prove that the index of every $m$-dimensional vector bundle over $B$ is equal to $m$ if $m \geq 2 dim{B}$. We also determine the smallest integer $k$ for which every $m$-dimensional vector bundle with $m\geq k$ is I-stable in the cases $B=FP^n$ and $B=S^n$.