Stable extendibility of normal bundles over lens spaces (Q1957144)

Let \(L^n(p)=S^{2n+1}/(\mathbb Z/p)\) be the \((2n+1)\)-dimensional standard lens space for an odd prime \(p\). For any \(\mathbb R\)-vector bundle \(\zeta\) over \(L^n(p)\), \(s(\zeta)\) denotes the maximal number \(m\) such that \(\zeta\) is stably extendible to \(L^m(p)\). In the paper under review, the author studies \(s(\nu_n^t(p))\) for the normal bundle \(\nu _n^t(p)\) of an immersion of \(L^n(p)\) in the Euclidean space \(\mathbb R^{2n+1+t}\). First the author shows that if \((p-1)/2\leq n \leq p\) for \(p\geq5\) and \(0\leq n \leq 5\) for \(p=3\), then \(s(\nu_n^t(p))=\infty\) for \(t\geq 2n+1\). To prove the theorem, the author shows that \(\nu_n^t(p)\), under these conditions, is stably equivalent to a bundle which is the Whitney sum of \(2\)-plane bundles and a trivial bundle. For the case of \(p=3\) or \(5\), a more explicit result is given. From the theorem, the author addresses the conjecture that \(s(\nu_n^t(p))=\infty\) if \(n\leq p\). Next the author shows that if \(n\geq 2p\) and \(t\leq 2n+2\), then \(s(\nu_n^t(p))<\infty\) for \(p\geq7\). Finally, the author studies actual values of \(s(\nu_n^t(p))\) for the case of \(s(\nu_n^t(p))<\infty\).