Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R -vector bundles over the ( 2 n + 1 ) -dimensional standard lens space L n ( p ) with odd prime p, focusing on the normal bundle ν n t ( p ) to an immersion of L n ( p ) in the Euclidean space R 2 n + 1 + t . We show several concrete cases in which ν n t ( p ) is stably extendible to L k ( p ) for any k with k ⩾ n , and in several cases we determine the exact value m for which ν n t ( p ) is stably extendible to L m ( p ) but not stably extendible to L m + 1 ( p ) .