Stable extendibility of the tangent bundles over lens spaces

Abstract

The purpose of this paper is to study the stable extendibility of the tangent bundle $\tau_n(p)$ of the $(2n+1)$-\hspace dimensional standard lens space $\mL^n(p)$ for odd prime $p$. We investigate the value of integer $m$ for which $\tau_n(p)$ is stably extendible to $\mL^m(p)$ but not stably extendible to $\mL^{m+1}(p)$, and in particular we completely determine $m$ for $p=5$ or $7$. A stable splitting of $\tau_n(p)$ and the stable extendibility of a Whitney sum of $\tau_n(p)$ are also discussed.